Calculus Examples

$y = x^{\frac{12}{11}}$ , $y = 10 x^{\frac{1}{11}}$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$x^{\frac{12}{11}} = 10 x^{\frac{1}{11}}$

Step 1.2

Solve $x^{\frac{12}{11}} = 10 x^{\frac{1}{11}}$ for $x$ .

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Step 1.2.1

Eliminate the fractional exponents by multiplying both exponents by the LCD.

${(x^{\frac{12}{11}})}^{11} = {(10 x^{\frac{1}{11}})}^{11}$

Step 1.2.2

Multiply the exponents in ${(x^{\frac{12}{11}})}^{11}$ .

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Step 1.2.2.1

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

$x^{\frac{12}{11} \cdot 11} = {(10 x^{\frac{1}{11}})}^{11}$

Step 1.2.2.2

Cancel the common factor of $11$ .

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Step 1.2.2.2.1

Cancel the common factor.

$x^{\frac{12}{11} \cdot 11} = {(10 x^{\frac{1}{11}})}^{11}$

Step 1.2.2.2.2

Rewrite the expression.

$x^{12} = {(10 x^{\frac{1}{11}})}^{11}$

Step 1.2.3

Simplify ${(10 x^{\frac{1}{11}})}^{11}$ .

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Step 1.2.3.1

Apply the product rule to $10 x^{\frac{1}{11}}$ .

$x^{12} = 10^{11} {(x^{\frac{1}{11}})}^{11}$

Step 1.2.3.2

Raise $10$ to the power of $11$ .

$x^{12} = 100000000000 {(x^{\frac{1}{11}})}^{11}$

Step 1.2.3.3

Multiply the exponents in ${(x^{\frac{1}{11}})}^{11}$ .

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Step 1.2.3.3.1

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

$x^{12} = 100000000000 x^{\frac{1}{11} \cdot 11}$

Step 1.2.3.3.2

Cancel the common factor of $11$ .

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Step 1.2.3.3.2.1

Cancel the common factor.

$x^{12} = 100000000000 x^{\frac{1}{11} \cdot 11}$

Step 1.2.3.3.2.2

Rewrite the expression.

$x^{12} = 100000000000 x^{1}$

Step 1.2.3.4

Simplify.

$x^{12} = 100000000000 x$

Step 1.2.4

Subtract $100000000000 x$ from both sides of the equation.

$x^{12} - 100000000000 x = 0$

Step 1.2.5

Factor $x$ out of $x^{12} - 100000000000 x$ .

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Step 1.2.5.1

Factor $x$ out of $x^{12}$ .

$x \cdot x^{11} - 100000000000 x = 0$

Step 1.2.5.2

Factor $x$ out of $- 100000000000 x$ .

$x \cdot x^{11} + x \cdot - 100000000000 = 0$

Step 1.2.5.3

Factor $x$ out of $x \cdot x^{11} + x \cdot - 100000000000$ .

$x (x^{11} - 100000000000) = 0$

Step 1.2.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x^{11} - 100000000000 = 0 + y = 10 x^{\frac{1}{11}}$

Step 1.2.7

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.8

Set $x^{11} - 100000000000$ equal to $0$ and solve for $x$ .

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Step 1.2.8.1

Set $x^{11} - 100000000000$ equal to $0$ .

$x^{11} - 100000000000 = 0$

Step 1.2.8.2

Solve $x^{11} - 100000000000 = 0$ for $x$ .

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Step 1.2.8.2.1

Add $100000000000$ to both sides of the equation.

$x^{11} = 100000000000$

Step 1.2.8.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[11]{100000000000}$

Step 1.2.8.2.3

Simplify $\sqrt[11]{100000000000}$ .

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Step 1.2.8.2.3.1

Rewrite $100000000000$ as $10^{11}$ .

$x = \sqrt[11]{10^{11}}$

Step 1.2.8.2.3.2

Pull terms out from under the radical, assuming real numbers.

$x = 10$

Step 1.2.9

The final solution is all the values that make $x (x^{11} - 100000000000) = 0$ true.

$x = 0, 10$

Step 1.3

Evaluate $y$ when $x = 0$ .

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Step 1.3.1

Substitute $0$ for $x$ .

$y = 10 {(0)}^{\frac{1}{11}}$

Step 1.3.2

Substitute $0$ for $x$ in $y = 10 {(0)}^{\frac{1}{11}}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = 10 \cdot 0^{\frac{1}{11}}$

Step 1.3.2.2

Simplify $10 \cdot 0^{\frac{1}{11}}$ .

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Step 1.3.2.2.1

Simplify the expression.

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Step 1.3.2.2.1.1

Rewrite $0$ as $0^{11}$ .

$y = 10 \cdot {(0^{11})}^{\frac{1}{11}}$

Step 1.3.2.2.1.2

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

$y = 10 \cdot 0^{11 (\frac{1}{11})}$

Step 1.3.2.2.2

Cancel the common factor of $11$ .

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Step 1.3.2.2.2.1

Cancel the common factor.

$y = 10 \cdot 0^{11 (\frac{1}{11})}$

Step 1.3.2.2.2.2

Rewrite the expression.

$y = 10 \cdot 0^{1}$

Step 1.3.2.2.3

Evaluate the exponent.

$y = 10 \cdot 0$

Step 1.3.2.2.4

Multiply $10$ by $0$ .

$y = 0$

Step 1.4

Evaluate $y$ when $x = 10$ .

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Step 1.4.1

Substitute $10$ for $x$ .

$y = 10 {(10)}^{\frac{1}{11}}$

Step 1.4.2

Substitute $10$ for $x$ in $y = 10 {(10)}^{\frac{1}{11}}$ and solve for $y$ .

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Step 1.4.2.1

Remove parentheses.

$y = 10 \cdot 10^{\frac{1}{11}}$

Step 1.4.2.2

Multiply $10$ by $10^{\frac{1}{11}}$ by adding the exponents.

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Step 1.4.2.2.1

Multiply $10$ by $10^{\frac{1}{11}}$ .

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Step 1.4.2.2.1.1

Raise $10$ to the power of $1$ .

$y = 10^{1} \cdot 10^{\frac{1}{11}}$

Step 1.4.2.2.1.2

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

$y = 10^{1 + \frac{1}{11}}$

Step 1.4.2.2.2

Write $1$ as a fraction with a common denominator.

$y = 10^{\frac{11}{11} + \frac{1}{11}}$

Step 1.4.2.2.3

Combine the numerators over the common denominator.

$y = 10^{\frac{11 + 1}{11}}$

Step 1.4.2.2.4

Add $11$ and $1$ .

$y = 10^{\frac{12}{11}}$

Step 1.5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, 0)$

$(10, 10^{\frac{12}{11}})$

$(0, 0)$

$(10, 10^{\frac{12}{11}})$

Step 2

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{0}^{10} 10 x^{\frac{1}{11}} d x - \int_{0}^{10} x^{\frac{12}{11}} d x$

Step 3

Integrate to find the area between $0$ and $10$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{0}^{10} 10 x^{\frac{1}{11}} - (x^{\frac{12}{11}}) d x$

Step 3.2

Multiply $- 1$ by $x^{\frac{12}{11}}$ .

$\int_{0}^{10} 10 x^{\frac{1}{11}} - x^{\frac{12}{11}} d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{0}^{10} 10 x^{\frac{1}{11}} d x + \int_{0}^{10} - x^{\frac{12}{11}} d x$

Step 3.4

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$10 \int_{0}^{10} x^{\frac{1}{11}} d x + \int_{0}^{10} - x^{\frac{12}{11}} d x$

Step 3.5

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

$10 (\frac{11}{12} x^{\frac{12}{11}}]_{0}^{10}) + \int_{0}^{10} - x^{\frac{12}{11}} d x$

Step 3.6

Combine $\frac{11}{12}$ and $x^{\frac{12}{11}}$ .

$10 (\frac{11 x^{\frac{12}{11}}}{12}]_{0}^{10}) + \int_{0}^{10} - x^{\frac{12}{11}} d x$

Step 3.7

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$10 (\frac{11 x^{\frac{12}{11}}}{12}]_{0}^{10}) - \int_{0}^{10} x^{\frac{12}{11}} d x$

Step 3.8

By the Power Rule, the integral of $x^{\frac{12}{11}}$ with respect to $x$ is $\frac{11}{23} x^{\frac{23}{11}}$ .

$10 (\frac{11 x^{\frac{12}{11}}}{12}]_{0}^{10}) - (\frac{11}{23} x^{\frac{23}{11}}]_{0}^{10})$

Step 3.9

Simplify the answer.

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Step 3.9.1

Combine $\frac{11}{23}$ and $x^{\frac{23}{11}}$ .

$10 (\frac{11 x^{\frac{12}{11}}}{12}]_{0}^{10}) - (\frac{11 x^{\frac{23}{11}}}{23}]_{0}^{10})$

Step 3.9.2

Substitute and simplify.

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Step 3.9.2.1

Evaluate $\frac{11 x^{\frac{12}{11}}}{12}$ at $10$ and at $0$ .

$10 ((\frac{11 \cdot 10^{\frac{12}{11}}}{12}) - \frac{11 \cdot 0^{\frac{12}{11}}}{12}) - (\frac{11 x^{\frac{23}{11}}}{23}]_{0}^{10})$

Step 3.9.2.2

Evaluate $\frac{11 x^{\frac{23}{11}}}{23}$ at $10$ and at $0$ .

$10 (\frac{11 \cdot 10^{\frac{12}{11}}}{12} - \frac{11 \cdot 0^{\frac{12}{11}}}{12}) - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3

Simplify.

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Step 3.9.2.3.1

Rewrite $0$ as $0^{11}$ .

$10 (\frac{11 \cdot 10^{\frac{12}{11}}}{12} - \frac{11 \cdot {(0^{11})}^{\frac{12}{11}}}{12}) - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.2

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

$10 (\frac{11 \cdot 10^{\frac{12}{11}}}{12} - \frac{11 \cdot 0^{11 (\frac{12}{11})}}{12}) - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.3

Cancel the common factor of $11$ .

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Step 3.9.2.3.3.1

Cancel the common factor.

$10 (\frac{11 \cdot 10^{\frac{12}{11}}}{12} - \frac{11 \cdot 0^{11 (\frac{12}{11})}}{12}) - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.3.2

Rewrite the expression.

$10 (\frac{11 \cdot 10^{\frac{12}{11}}}{12} - \frac{11 \cdot 0^{12}}{12}) - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.4

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

$10 (\frac{11 \cdot 10^{\frac{12}{11}}}{12} - \frac{11 \cdot 0}{12}) - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.5

Multiply $11$ by $0$ .

$10 (\frac{11 \cdot 10^{\frac{12}{11}}}{12} - \frac{0}{12}) - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.6

Cancel the common factor of $0$ and $12$ .

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Step 3.9.2.3.6.1

Factor $12$ out of $0$ .

$10 (\frac{11 \cdot 10^{\frac{12}{11}}}{12} - \frac{12 (0)}{12}) - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.6.2

Cancel the common factors.

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Step 3.9.2.3.6.2.1

Factor $12$ out of $12$ .

$10 (\frac{11 \cdot 10^{\frac{12}{11}}}{12} - \frac{12 \cdot 0}{12 \cdot 1}) - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.6.2.2

Cancel the common factor.

$10 (\frac{11 \cdot 10^{\frac{12}{11}}}{12} - \frac{12 \cdot 0}{12 \cdot 1}) - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.6.2.3

Rewrite the expression.

$10 (\frac{11 \cdot 10^{\frac{12}{11}}}{12} - \frac{0}{1}) - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.6.2.4

Divide $0$ by $1$ .

$10 (\frac{11 \cdot 10^{\frac{12}{11}}}{12} - 0) - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.7

Multiply $- 1$ by $0$ .

$10 (\frac{11 \cdot 10^{\frac{12}{11}}}{12} + 0) - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.8

Add $\frac{11 \cdot 10^{\frac{12}{11}}}{12}$ and $0$ .

$10 \frac{11 \cdot 10^{\frac{12}{11}}}{12} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.9

Combine $10$ and $\frac{11 \cdot 10^{\frac{12}{11}}}{12}$ .

$\frac{10 (11 \cdot 10^{\frac{12}{11}})}{12} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.10

Multiply $11$ by $10$ .

$\frac{110 \cdot 10^{\frac{12}{11}}}{12} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.11

Factor $2$ out of $110 \cdot 10^{\frac{12}{11}}$ .

$\frac{2 (55 \cdot 10^{\frac{12}{11}})}{12} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.12

Cancel the common factors.

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Step 3.9.2.3.12.1

Factor $2$ out of $12$ .

$\frac{2 (55 \cdot 10^{\frac{12}{11}})}{2 (6)} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.12.2

Cancel the common factor.

$\frac{2 (55 \cdot 10^{\frac{12}{11}})}{2 \cdot 6} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.12.3

Rewrite the expression.

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{\frac{23}{11}}}{23})$

Step 3.9.2.3.13

Rewrite $0$ as $0^{11}$ .

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot {(0^{11})}^{\frac{23}{11}}}{23})$

Step 3.9.2.3.14

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

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{11 (\frac{23}{11})}}{23})$

Step 3.9.2.3.15

Cancel the common factor of $11$ .

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Step 3.9.2.3.15.1

Cancel the common factor.

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{11 (\frac{23}{11})}}{23})$

Step 3.9.2.3.15.2

Rewrite the expression.

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0^{23}}{23})$

Step 3.9.2.3.16

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

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{11 \cdot 0}{23})$

Step 3.9.2.3.17

Multiply $11$ by $0$ .

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{0}{23})$

Step 3.9.2.3.18

Cancel the common factor of $0$ and $23$ .

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Step 3.9.2.3.18.1

Factor $23$ out of $0$ .

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{23 (0)}{23})$

Step 3.9.2.3.18.2

Cancel the common factors.

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Step 3.9.2.3.18.2.1

Factor $23$ out of $23$ .

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{23 \cdot 0}{23 \cdot 1})$

Step 3.9.2.3.18.2.2

Cancel the common factor.

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{23 \cdot 0}{23 \cdot 1})$

Step 3.9.2.3.18.2.3

Rewrite the expression.

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - \frac{0}{1})$

Step 3.9.2.3.18.2.4

Divide $0$ by $1$ .

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} - 0)$

Step 3.9.2.3.19

Multiply $- 1$ by $0$ .

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} - (\frac{11 \cdot 10^{\frac{23}{11}}}{23} + 0)$

Step 3.9.2.3.20

Add $\frac{11 \cdot 10^{\frac{23}{11}}}{23}$ and $0$ .

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} - \frac{11 \cdot 10^{\frac{23}{11}}}{23}$

Step 3.9.2.3.21

To write $\frac{55 \cdot 10^{\frac{12}{11}}}{6}$ as a fraction with a common denominator, multiply by $\frac{23}{23}$ .

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} \cdot \frac{23}{23} - \frac{11 \cdot 10^{\frac{23}{11}}}{23}$

Step 3.9.2.3.22

To write $- \frac{11 \cdot 10^{\frac{23}{11}}}{23}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{55 \cdot 10^{\frac{12}{11}}}{6} \cdot \frac{23}{23} - \frac{11 \cdot 10^{\frac{23}{11}}}{23} \cdot \frac{6}{6}$

Step 3.9.2.3.23

Write each expression with a common denominator of $138$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.9.2.3.23.1

Multiply $\frac{55 \cdot 10^{\frac{12}{11}}}{6}$ by $\frac{23}{23}$ .

$\frac{55 \cdot 10^{\frac{12}{11}} \cdot 23}{6 \cdot 23} - \frac{11 \cdot 10^{\frac{23}{11}}}{23} \cdot \frac{6}{6}$

Step 3.9.2.3.23.2

Multiply $6$ by $23$ .

$\frac{55 \cdot 10^{\frac{12}{11}} \cdot 23}{138} - \frac{11 \cdot 10^{\frac{23}{11}}}{23} \cdot \frac{6}{6}$

Step 3.9.2.3.23.3

Multiply $\frac{11 \cdot 10^{\frac{23}{11}}}{23}$ by $\frac{6}{6}$ .

$\frac{55 \cdot 10^{\frac{12}{11}} \cdot 23}{138} - \frac{11 \cdot 10^{\frac{23}{11}} \cdot 6}{23 \cdot 6}$

Step 3.9.2.3.23.4

Multiply $23$ by $6$ .

$\frac{55 \cdot 10^{\frac{12}{11}} \cdot 23}{138} - \frac{11 \cdot 10^{\frac{23}{11}} \cdot 6}{138}$

Step 3.9.2.3.24

Combine the numerators over the common denominator.

$\frac{55 \cdot 10^{\frac{12}{11}} \cdot 23 - 11 \cdot 10^{\frac{23}{11}} \cdot 6}{138}$

Step 3.9.2.3.25

Multiply $23$ by $55$ .

$\frac{1265 \cdot 10^{\frac{12}{11}} - 11 \cdot 10^{\frac{23}{11}} \cdot 6}{138}$

Step 3.9.2.3.26

Multiply $6$ by $- 11$ .

$\frac{1265 \cdot 10^{\frac{12}{11}} - 66 \cdot 10^{\frac{23}{11}}}{138}$

Step 4