Calculus Examples

$\int_{0}^{1} {(x^{2} - 1)}^{4} x^{3} d x$

Step 1

Simplify.

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Step 1.1

Use the Binomial Theorem.

$\int_{0}^{1} ({(x^{2})}^{4} + 4 {(x^{2})}^{3} \cdot - 1 + 6 {(x^{2})}^{2} {(- 1)}^{2} + 4 x^{2} {(- 1)}^{3} + {(- 1)}^{4}) x^{3} d x$

Step 1.2

Simplify each term.

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Step 1.2.1

Multiply the exponents in ${(x^{2})}^{4}$ .

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Step 1.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int_{0}^{1} (x^{2 \cdot 4} + 4 {(x^{2})}^{3} \cdot - 1 + 6 {(x^{2})}^{2} {(- 1)}^{2} + 4 x^{2} {(- 1)}^{3} + {(- 1)}^{4}) x^{3} d x$

Step 1.2.1.2

Multiply $2$ by $4$ .

$\int_{0}^{1} (x^{8} + 4 {(x^{2})}^{3} \cdot - 1 + 6 {(x^{2})}^{2} {(- 1)}^{2} + 4 x^{2} {(- 1)}^{3} + {(- 1)}^{4}) x^{3} d x$

Step 1.2.2

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 1.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int_{0}^{1} (x^{8} + 4 x^{2 \cdot 3} \cdot - 1 + 6 {(x^{2})}^{2} {(- 1)}^{2} + 4 x^{2} {(- 1)}^{3} + {(- 1)}^{4}) x^{3} d x$

Step 1.2.2.2

Multiply $2$ by $3$ .

$\int_{0}^{1} (x^{8} + 4 x^{6} \cdot - 1 + 6 {(x^{2})}^{2} {(- 1)}^{2} + 4 x^{2} {(- 1)}^{3} + {(- 1)}^{4}) x^{3} d x$

Step 1.2.3

Multiply $- 1$ by $4$ .

$\int_{0}^{1} (x^{8} - 4 x^{6} + 6 {(x^{2})}^{2} {(- 1)}^{2} + 4 x^{2} {(- 1)}^{3} + {(- 1)}^{4}) x^{3} d x$

Step 1.2.4

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.2.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int_{0}^{1} (x^{8} - 4 x^{6} + 6 x^{2 \cdot 2} {(- 1)}^{2} + 4 x^{2} {(- 1)}^{3} + {(- 1)}^{4}) x^{3} d x$

Step 1.2.4.2

Multiply $2$ by $2$ .

$\int_{0}^{1} (x^{8} - 4 x^{6} + 6 x^{4} {(- 1)}^{2} + 4 x^{2} {(- 1)}^{3} + {(- 1)}^{4}) x^{3} d x$

Step 1.2.5

Raise $- 1$ to the power of $2$ .

$\int_{0}^{1} (x^{8} - 4 x^{6} + 6 x^{4} \cdot 1 + 4 x^{2} {(- 1)}^{3} + {(- 1)}^{4}) x^{3} d x$

Step 1.2.6

Multiply $6$ by $1$ .

$\int_{0}^{1} (x^{8} - 4 x^{6} + 6 x^{4} + 4 x^{2} {(- 1)}^{3} + {(- 1)}^{4}) x^{3} d x$

Step 1.2.7

Raise $- 1$ to the power of $3$ .

$\int_{0}^{1} (x^{8} - 4 x^{6} + 6 x^{4} + 4 x^{2} \cdot - 1 + {(- 1)}^{4}) x^{3} d x$

Step 1.2.8

Multiply $- 1$ by $4$ .

$\int_{0}^{1} (x^{8} - 4 x^{6} + 6 x^{4} - 4 x^{2} + {(- 1)}^{4}) x^{3} d x$

Step 1.2.9

Raise $- 1$ to the power of $4$ .

$\int_{0}^{1} (x^{8} - 4 x^{6} + 6 x^{4} - 4 x^{2} + 1) x^{3} d x$

Step 1.3

Apply the distributive property.

$\int_{0}^{1} x^{8} x^{3} - 4 x^{6} x^{3} + 6 x^{4} x^{3} - 4 x^{2} x^{3} + 1 x^{3} d x$

Step 1.4

Simplify.

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Step 1.4.1

Multiply $x^{8}$ by $x^{3}$ by adding the exponents.

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Step 1.4.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{0}^{1} x^{8 + 3} - 4 x^{6} x^{3} + 6 x^{4} x^{3} - 4 x^{2} x^{3} + 1 x^{3} d x$

Step 1.4.1.2

Add $8$ and $3$ .

$\int_{0}^{1} x^{11} - 4 x^{6} x^{3} + 6 x^{4} x^{3} - 4 x^{2} x^{3} + 1 x^{3} d x$

Step 1.4.2

Multiply $x^{6}$ by $x^{3}$ by adding the exponents.

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Step 1.4.2.1

Move $x^{3}$ .

$\int_{0}^{1} x^{11} - 4 (x^{3} x^{6}) + 6 x^{4} x^{3} - 4 x^{2} x^{3} + 1 x^{3} d x$

Step 1.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{0}^{1} x^{11} - 4 x^{3 + 6} + 6 x^{4} x^{3} - 4 x^{2} x^{3} + 1 x^{3} d x$

Step 1.4.2.3

Add $3$ and $6$ .

$\int_{0}^{1} x^{11} - 4 x^{9} + 6 x^{4} x^{3} - 4 x^{2} x^{3} + 1 x^{3} d x$

Step 1.4.3

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 1.4.3.1

Move $x^{3}$ .

$\int_{0}^{1} x^{11} - 4 x^{9} + 6 (x^{3} x^{4}) - 4 x^{2} x^{3} + 1 x^{3} d x$

Step 1.4.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{0}^{1} x^{11} - 4 x^{9} + 6 x^{3 + 4} - 4 x^{2} x^{3} + 1 x^{3} d x$

Step 1.4.3.3

Add $3$ and $4$ .

$\int_{0}^{1} x^{11} - 4 x^{9} + 6 x^{7} - 4 x^{2} x^{3} + 1 x^{3} d x$

Step 1.4.4

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 1.4.4.1

Move $x^{3}$ .

$\int_{0}^{1} x^{11} - 4 x^{9} + 6 x^{7} - 4 (x^{3} x^{2}) + 1 x^{3} d x$

Step 1.4.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{0}^{1} x^{11} - 4 x^{9} + 6 x^{7} - 4 x^{3 + 2} + 1 x^{3} d x$

Step 1.4.4.3

Add $3$ and $2$ .

$\int_{0}^{1} x^{11} - 4 x^{9} + 6 x^{7} - 4 x^{5} + 1 x^{3} d x$

Step 1.4.5

Multiply $x^{3}$ by $1$ .

$\int_{0}^{1} x^{11} - 4 x^{9} + 6 x^{7} - 4 x^{5} + x^{3} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{1} x^{11} d x + \int_{0}^{1} - 4 x^{9} d x + \int_{0}^{1} 6 x^{7} d x + \int_{0}^{1} - 4 x^{5} d x + \int_{0}^{1} x^{3} d x$

Step 3

By the Power Rule, the integral of $x^{11}$ with respect to $x$ is $\frac{1}{12} x^{12}$ .

$\frac{1}{12} x^{12}]_{0}^{1} + \int_{0}^{1} - 4 x^{9} d x + \int_{0}^{1} 6 x^{7} d x + \int_{0}^{1} - 4 x^{5} d x + \int_{0}^{1} x^{3} d x$

Step 4

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$\frac{1}{12} x^{12}]_{0}^{1} - 4 \int_{0}^{1} x^{9} d x + \int_{0}^{1} 6 x^{7} d x + \int_{0}^{1} - 4 x^{5} d x + \int_{0}^{1} x^{3} d x$

Step 5

By the Power Rule, the integral of $x^{9}$ with respect to $x$ is $\frac{1}{10} x^{10}$ .

$\frac{1}{12} x^{12}]_{0}^{1} - 4 (\frac{1}{10} x^{10}]_{0}^{1}) + \int_{0}^{1} 6 x^{7} d x + \int_{0}^{1} - 4 x^{5} d x + \int_{0}^{1} x^{3} d x$

Step 6

Combine $\frac{1}{10}$ and $x^{10}$ .

$\frac{1}{12} x^{12}]_{0}^{1} - 4 (\frac{x^{10}}{10}]_{0}^{1}) + \int_{0}^{1} 6 x^{7} d x + \int_{0}^{1} - 4 x^{5} d x + \int_{0}^{1} x^{3} d x$

Step 7

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$\frac{1}{12} x^{12}]_{0}^{1} - 4 (\frac{x^{10}}{10}]_{0}^{1}) + 6 \int_{0}^{1} x^{7} d x + \int_{0}^{1} - 4 x^{5} d x + \int_{0}^{1} x^{3} d x$

Step 8

By the Power Rule, the integral of $x^{7}$ with respect to $x$ is $\frac{1}{8} x^{8}$ .

$\frac{1}{12} x^{12}]_{0}^{1} - 4 (\frac{x^{10}}{10}]_{0}^{1}) + 6 (\frac{1}{8} x^{8}]_{0}^{1}) + \int_{0}^{1} - 4 x^{5} d x + \int_{0}^{1} x^{3} d x$

Step 9

Combine $\frac{1}{8}$ and $x^{8}$ .

$\frac{1}{12} x^{12}]_{0}^{1} - 4 (\frac{x^{10}}{10}]_{0}^{1}) + 6 (\frac{x^{8}}{8}]_{0}^{1}) + \int_{0}^{1} - 4 x^{5} d x + \int_{0}^{1} x^{3} d x$

Step 10

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$\frac{1}{12} x^{12}]_{0}^{1} - 4 (\frac{x^{10}}{10}]_{0}^{1}) + 6 (\frac{x^{8}}{8}]_{0}^{1}) - 4 \int_{0}^{1} x^{5} d x + \int_{0}^{1} x^{3} d x$

Step 11

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$\frac{1}{12} x^{12}]_{0}^{1} - 4 (\frac{x^{10}}{10}]_{0}^{1}) + 6 (\frac{x^{8}}{8}]_{0}^{1}) - 4 (\frac{1}{6} x^{6}]_{0}^{1}) + \int_{0}^{1} x^{3} d x$

Step 12

Combine $\frac{1}{6}$ and $x^{6}$ .

$\frac{1}{12} x^{12}]_{0}^{1} - 4 (\frac{x^{10}}{10}]_{0}^{1}) + 6 (\frac{x^{8}}{8}]_{0}^{1}) - 4 (\frac{x^{6}}{6}]_{0}^{1}) + \int_{0}^{1} x^{3} d x$

Step 13

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{1}{12} x^{12}]_{0}^{1} - 4 (\frac{x^{10}}{10}]_{0}^{1}) + 6 (\frac{x^{8}}{8}]_{0}^{1}) - 4 (\frac{x^{6}}{6}]_{0}^{1}) + \frac{1}{4} x^{4}]_{0}^{1}$

Step 14

Simplify the answer.

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Step 14.1

Combine $\frac{1}{12} x^{12}]_{0}^{1}$ and $\frac{1}{4} x^{4}]_{0}^{1}$ .

$\frac{1}{12} x^{12} + \frac{1}{4} x^{4}]_{0}^{1} - 4 (\frac{x^{10}}{10}]_{0}^{1}) + 6 (\frac{x^{8}}{8}]_{0}^{1}) - 4 (\frac{x^{6}}{6}]_{0}^{1})$

Step 14.2

Substitute and simplify.

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Step 14.2.1

Evaluate $\frac{1}{12} x^{12} + \frac{1}{4} x^{4}$ at $1$ and at $0$ .

$(\frac{1}{12} \cdot 1^{12} + \frac{1}{4} \cdot 1^{4}) - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{x^{10}}{10}]_{0}^{1}) + 6 (\frac{x^{8}}{8}]_{0}^{1}) - 4 (\frac{x^{6}}{6}]_{0}^{1})$

Step 14.2.2

Evaluate $\frac{x^{10}}{10}$ at $1$ and at $0$ .

$(\frac{1}{12} \cdot 1^{12} + \frac{1}{4} \cdot 1^{4}) - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10}) + 6 (\frac{x^{8}}{8}]_{0}^{1}) - 4 (\frac{x^{6}}{6}]_{0}^{1})$

Step 14.2.3

Evaluate $\frac{x^{8}}{8}$ at $1$ and at $0$ .

$(\frac{1}{12} \cdot 1^{12} + \frac{1}{4} \cdot 1^{4}) - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 (\frac{x^{6}}{6}]_{0}^{1})$

Step 14.2.4

Evaluate $\frac{x^{6}}{6}$ at $1$ and at $0$ .

Step 14.2.5

Simplify.

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Step 14.2.5.1

One to any power is one.

$\frac{1}{12} \cdot 1 + \frac{1}{4} \cdot 1^{4} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.2

Multiply $\frac{1}{12}$ by $1$ .

$\frac{1}{12} + \frac{1}{4} \cdot 1^{4} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.3

One to any power is one.

$\frac{1}{12} + \frac{1}{4} \cdot 1 - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.4

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{12} + \frac{1}{4} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.5

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{12} + \frac{1}{4} \cdot \frac{3}{3} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.6

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.2.5.6.1

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$\frac{1}{12} + \frac{3}{4 \cdot 3} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.6.2

Multiply $4$ by $3$ .

$\frac{1}{12} + \frac{3}{12} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.7

Combine the numerators over the common denominator.

$\frac{1 + 3}{12} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.8

Add $1$ and $3$ .

$\frac{4}{12} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.9

Cancel the common factor of $4$ and $12$ .

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Step 14.2.5.9.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{12} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.9.2

Cancel the common factors.

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Step 14.2.5.9.2.1

Factor $4$ out of $12$ .

$\frac{4 \cdot 1}{4 \cdot 3} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.9.2.2

Cancel the common factor.

Step 14.2.5.9.2.3

Rewrite the expression.

$\frac{1}{3} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.10

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3} - (\frac{1}{12} \cdot 0 + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.11

Multiply $\frac{1}{12}$ by $0$ .

$\frac{1}{3} - (0 + \frac{1}{4} \cdot 0^{4}) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.12

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3} - (0 + \frac{1}{4} \cdot 0) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.13

Multiply $\frac{1}{4}$ by $0$ .

$\frac{1}{3} - (0 + 0) - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.14

Add $0$ and $0$ .

$\frac{1}{3} - 0 - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.15

Multiply $- 1$ by $0$ .

$\frac{1}{3} + 0 - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.16

Add $\frac{1}{3}$ and $0$ .

$\frac{1}{3} - 4 (\frac{1^{10}}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.17

One to any power is one.

$\frac{1}{3} - 4 (\frac{1}{10} - \frac{0^{10}}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.18

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3} - 4 (\frac{1}{10} - \frac{0}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.19

Cancel the common factor of $0$ and $10$ .

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Step 14.2.5.19.1

Factor $10$ out of $0$ .

$\frac{1}{3} - 4 (\frac{1}{10} - \frac{10 (0)}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.19.2

Cancel the common factors.

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Step 14.2.5.19.2.1

Factor $10$ out of $10$ .

$\frac{1}{3} - 4 (\frac{1}{10} - \frac{10 \cdot 0}{10 \cdot 1}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.19.2.2

Cancel the common factor.

$\frac{1}{3} - 4 (\frac{1}{10} - \frac{10 \cdot 0}{10 \cdot 1}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.19.2.3

Rewrite the expression.

$\frac{1}{3} - 4 (\frac{1}{10} - \frac{0}{1}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.19.2.4

Divide $0$ by $1$ .

$\frac{1}{3} - 4 (\frac{1}{10} - 0) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.20

Multiply $- 1$ by $0$ .

$\frac{1}{3} - 4 (\frac{1}{10} + 0) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.21

Add $\frac{1}{10}$ and $0$ .

$\frac{1}{3} - 4 (\frac{1}{10}) + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.22

Combine $- 4$ and $\frac{1}{10}$ .

$\frac{1}{3} + \frac{- 4}{10} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.23

Cancel the common factor of $- 4$ and $10$ .

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Step 14.2.5.23.1

Factor $2$ out of $- 4$ .

$\frac{1}{3} + \frac{2 (- 2)}{10} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.23.2

Cancel the common factors.

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Step 14.2.5.23.2.1

Factor $2$ out of $10$ .

$\frac{1}{3} + \frac{2 \cdot - 2}{2 \cdot 5} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.23.2.2

Cancel the common factor.

$\frac{1}{3} + \frac{2 \cdot - 2}{2 \cdot 5} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.23.2.3

Rewrite the expression.

$\frac{1}{3} + \frac{- 2}{5} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.24

Move the negative in front of the fraction.

$\frac{1}{3} - \frac{2}{5} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.25

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1}{3} \cdot \frac{5}{5} - \frac{2}{5} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.26

To write $- \frac{2}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} \cdot \frac{5}{5} - \frac{2}{5} \cdot \frac{3}{3} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.27

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.2.5.27.1

Multiply $\frac{1}{3}$ by $\frac{5}{5}$ .

$\frac{5}{3 \cdot 5} - \frac{2}{5} \cdot \frac{3}{3} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.27.2

Multiply $3$ by $5$ .

$\frac{5}{15} - \frac{2}{5} \cdot \frac{3}{3} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.27.3

Multiply $\frac{2}{5}$ by $\frac{3}{3}$ .

$\frac{5}{15} - \frac{2 \cdot 3}{5 \cdot 3} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.27.4

Multiply $5$ by $3$ .

$\frac{5}{15} - \frac{2 \cdot 3}{15} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.28

Combine the numerators over the common denominator.

$\frac{5 - 2 \cdot 3}{15} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.29

Simplify the numerator.

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Step 14.2.5.29.1

Multiply $- 2$ by $3$ .

$\frac{5 - 6}{15} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.29.2

Subtract $6$ from $5$ .

$\frac{- 1}{15} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.30

Move the negative in front of the fraction.

$- \frac{1}{15} + 6 (\frac{1^{8}}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.31

One to any power is one.

$- \frac{1}{15} + 6 (\frac{1}{8} - \frac{0^{8}}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.32

Raising $0$ to any positive power yields $0$ .

$- \frac{1}{15} + 6 (\frac{1}{8} - \frac{0}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.33

Cancel the common factor of $0$ and $8$ .

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Step 14.2.5.33.1

Factor $8$ out of $0$ .

$- \frac{1}{15} + 6 (\frac{1}{8} - \frac{8 (0)}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.33.2

Cancel the common factors.

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Step 14.2.5.33.2.1

Factor $8$ out of $8$ .

$- \frac{1}{15} + 6 (\frac{1}{8} - \frac{8 \cdot 0}{8 \cdot 1}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.33.2.2

Cancel the common factor.

$- \frac{1}{15} + 6 (\frac{1}{8} - \frac{8 \cdot 0}{8 \cdot 1}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.33.2.3

Rewrite the expression.

$- \frac{1}{15} + 6 (\frac{1}{8} - \frac{0}{1}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.33.2.4

Divide $0$ by $1$ .

$- \frac{1}{15} + 6 (\frac{1}{8} - 0) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.34

Multiply $- 1$ by $0$ .

$- \frac{1}{15} + 6 (\frac{1}{8} + 0) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.35

Add $\frac{1}{8}$ and $0$ .

$- \frac{1}{15} + 6 (\frac{1}{8}) - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.36

Combine $6$ and $\frac{1}{8}$ .

$- \frac{1}{15} + \frac{6}{8} - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.37

Cancel the common factor of $6$ and $8$ .

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Step 14.2.5.37.1

Factor $2$ out of $6$ .

$- \frac{1}{15} + \frac{2 (3)}{8} - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.37.2

Cancel the common factors.

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Step 14.2.5.37.2.1

Factor $2$ out of $8$ .

$- \frac{1}{15} + \frac{2 \cdot 3}{2 \cdot 4} - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.37.2.2

Cancel the common factor.

$- \frac{1}{15} + \frac{2 \cdot 3}{2 \cdot 4} - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.37.2.3

Rewrite the expression.

$- \frac{1}{15} + \frac{3}{4} - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.38

To write $- \frac{1}{15}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{1}{15} \cdot \frac{4}{4} + \frac{3}{4} - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.39

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$- \frac{1}{15} \cdot \frac{4}{4} + \frac{3}{4} \cdot \frac{15}{15} - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.40

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.2.5.40.1

Multiply $\frac{1}{15}$ by $\frac{4}{4}$ .

$- \frac{4}{15 \cdot 4} + \frac{3}{4} \cdot \frac{15}{15} - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.40.2

Multiply $15$ by $4$ .

$- \frac{4}{60} + \frac{3}{4} \cdot \frac{15}{15} - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.40.3

Multiply $\frac{3}{4}$ by $\frac{15}{15}$ .

$- \frac{4}{60} + \frac{3 \cdot 15}{4 \cdot 15} - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.40.4

Multiply $4$ by $15$ .

$- \frac{4}{60} + \frac{3 \cdot 15}{60} - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.41

Combine the numerators over the common denominator.

$\frac{- 4 + 3 \cdot 15}{60} - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.42

Simplify the numerator.

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Step 14.2.5.42.1

Multiply $3$ by $15$ .

$\frac{- 4 + 45}{60} - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.42.2

Add $- 4$ and $45$ .

$\frac{41}{60} - 4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6})$

Step 14.2.5.43

One to any power is one.

$\frac{41}{60} - 4 (\frac{1}{6} - \frac{0^{6}}{6})$

Step 14.2.5.44

Raising $0$ to any positive power yields $0$ .

$\frac{41}{60} - 4 (\frac{1}{6} - \frac{0}{6})$

Step 14.2.5.45

Cancel the common factor of $0$ and $6$ .

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Step 14.2.5.45.1

Factor $6$ out of $0$ .

$\frac{41}{60} - 4 (\frac{1}{6} - \frac{6 (0)}{6})$

Step 14.2.5.45.2

Cancel the common factors.

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Step 14.2.5.45.2.1

Factor $6$ out of $6$ .

$\frac{41}{60} - 4 (\frac{1}{6} - \frac{6 \cdot 0}{6 \cdot 1})$

Step 14.2.5.45.2.2

Cancel the common factor.

$\frac{41}{60} - 4 (\frac{1}{6} - \frac{6 \cdot 0}{6 \cdot 1})$

Step 14.2.5.45.2.3

Rewrite the expression.

$\frac{41}{60} - 4 (\frac{1}{6} - \frac{0}{1})$

Step 14.2.5.45.2.4

Divide $0$ by $1$ .

$\frac{41}{60} - 4 (\frac{1}{6} - 0)$

Step 14.2.5.46

Multiply $- 1$ by $0$ .

$\frac{41}{60} - 4 (\frac{1}{6} + 0)$

Step 14.2.5.47

Add $\frac{1}{6}$ and $0$ .

$\frac{41}{60} - 4 (\frac{1}{6})$

Step 14.2.5.48

Combine $- 4$ and $\frac{1}{6}$ .

$\frac{41}{60} + \frac{- 4}{6}$

Step 14.2.5.49

Cancel the common factor of $- 4$ and $6$ .

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Step 14.2.5.49.1

Factor $2$ out of $- 4$ .

$\frac{41}{60} + \frac{2 (- 2)}{6}$

Step 14.2.5.49.2

Cancel the common factors.

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Step 14.2.5.49.2.1

Factor $2$ out of $6$ .

$\frac{41}{60} + \frac{2 \cdot - 2}{2 \cdot 3}$

Step 14.2.5.49.2.2

Cancel the common factor.

$\frac{41}{60} + \frac{2 \cdot - 2}{2 \cdot 3}$

Step 14.2.5.49.2.3

Rewrite the expression.

$\frac{41}{60} + \frac{- 2}{3}$

Step 14.2.5.50

Move the negative in front of the fraction.

$\frac{41}{60} - \frac{2}{3}$

Step 14.2.5.51

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$\frac{41}{60} - \frac{2}{3} \cdot \frac{20}{20}$

Step 14.2.5.52

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.2.5.52.1

Multiply $\frac{2}{3}$ by $\frac{20}{20}$ .

$\frac{41}{60} - \frac{2 \cdot 20}{3 \cdot 20}$

Step 14.2.5.52.2

Multiply $3$ by $20$ .

$\frac{41}{60} - \frac{2 \cdot 20}{60}$

Step 14.2.5.53

Combine the numerators over the common denominator.

$\frac{41 - 2 \cdot 20}{60}$

Step 14.2.5.54

Simplify the numerator.

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Step 14.2.5.54.1

Multiply $- 2$ by $20$ .

$\frac{41 - 40}{60}$

Step 14.2.5.54.2

Subtract $40$ from $41$ .

$\frac{1}{60}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{60}$

Decimal Form:

$0.01\overline{6}$

Step 16