Basic Math Examples

$\frac{\frac{3}{5} \cdot ({(a b)}^{3} b^{3} a^{- 5})}{3 b^{- 3} a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1

Factor each term.

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Step 1.1

Remove unnecessary parentheses.

$\frac{\frac{3}{5} \cdot {(a b)}^{3} b^{3} a^{- 5}}{3 b^{- 3} a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.2

Factor.

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Step 1.2.1

Apply the product rule to $a b$ .

$\frac{\frac{3}{5} \cdot (a^{3} b^{3}) b^{3} a^{- 5}}{3 b^{- 3} a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.2.2

Remove unnecessary parentheses.

$\frac{\frac{3}{5} \cdot a^{3} b^{3} b^{3} a^{- 5}}{3 b^{- 3} a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.3

Combine exponents.

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Step 1.3.1

Combine $\frac{3}{5}$ and $a^{3}$ .

$\frac{\frac{3 a^{3}}{5} b^{3} b^{3} a^{- 5}}{3 b^{- 3} a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.3.2

Combine $\frac{3 a^{3}}{5}$ and $b^{3}$ .

$\frac{\frac{3 a^{3} b^{3}}{5} b^{3} a^{- 5}}{3 b^{- 3} a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.3.3

Combine $\frac{3 a^{3} b^{3}}{5}$ and $b^{3}$ .

$\frac{\frac{3 a^{3} b^{3} b^{3}}{5} a^{- 5}}{3 b^{- 3} a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.3.4

Multiply $b^{3}$ by $b^{3}$ by adding the exponents.

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Step 1.3.4.1

Move $b^{3}$ .

$\frac{\frac{3 a^{3} (b^{3} b^{3})}{5} a^{- 5}}{3 b^{- 3} a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.3.4.2

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

$\frac{\frac{3 a^{3} b^{3 + 3}}{5} a^{- 5}}{3 b^{- 3} a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.3.4.3

Add $3$ and $3$ .

$\frac{\frac{3 a^{3} b^{6}}{5} a^{- 5}}{3 b^{- 3} a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.3.5

Combine $\frac{3 a^{3} b^{6}}{5}$ and $a^{- 5}$ .

$\frac{\frac{3 a^{3} b^{6} a^{- 5}}{5}}{3 b^{- 3} a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.3.6

Multiply $a^{3}$ by $a^{- 5}$ by adding the exponents.

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Step 1.3.6.1

Move $a^{- 5}$ .

$\frac{\frac{3 (a^{- 5} a^{3}) b^{6}}{5}}{3 b^{- 3} a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.3.6.2

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

$\frac{\frac{3 a^{- 5 + 3} b^{6}}{5}}{3 b^{- 3} a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.3.6.3

Add $- 5$ and $3$ .

$\frac{\frac{3 a^{- 2} b^{6}}{5}}{3 b^{- 3} a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.4

Move $b^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{\frac{3 a^{- 2} b^{6}}{5} b^{3}}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{3 \frac{1}{a^{2}} b^{6}}{5} b^{3}}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.5.2

Combine exponents.

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Step 1.5.2.1

Combine $3$ and $\frac{1}{a^{2}}$ .

$\frac{\frac{\frac{3}{a^{2}} b^{6}}{5} b^{3}}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.5.2.2

Combine $\frac{3}{a^{2}}$ and $b^{6}$ .

$\frac{\frac{\frac{3 b^{6}}{a^{2}}}{5} b^{3}}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.6

Combine $\frac{\frac{3 b^{6}}{a^{2}}}{5}$ and $b^{3}$ .

$\frac{\frac{\frac{3 b^{6}}{a^{2}} b^{3}}{5}}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.7

Combine $\frac{3 b^{6}}{a^{2}}$ and $b^{3}$ .

$\frac{\frac{\frac{3 b^{6} b^{3}}{a^{2}}}{5}}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.8

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

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Step 1.8.1

Move $b^{3}$ .

$\frac{\frac{\frac{3 (b^{3} b^{6})}{a^{2}}}{5}}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.8.2

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

$\frac{\frac{\frac{3 b^{3 + 6}}{a^{2}}}{5}}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.8.3

Add $3$ and $6$ .

$\frac{\frac{\frac{3 b^{9}}{a^{2}}}{5}}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.9

Simplify the numerator.

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Step 1.9.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{\frac{3 b^{9}}{a^{2}} \cdot \frac{1}{5}}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.9.2

Combine.

$\frac{\frac{3 b^{9} \cdot 1}{a^{2} \cdot 5}}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.9.3

Multiply $3$ by $1$ .

$\frac{\frac{3 b^{9}}{a^{2} \cdot 5}}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.9.4

Move $5$ to the left of $a^{2}$ .

$\frac{\frac{3 b^{9}}{5 \cdot a^{2}}}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.9.5

Multiply $5$ by $a^{2}$ .

$\frac{\frac{3 b^{9}}{5 a^{2}}}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.10

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 b^{9}}{5 a^{2}} \cdot \frac{1}{3 a^{2}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.11

Combine.

$\frac{3 b^{9} \cdot 1}{5 a^{2} (3 a^{2})} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.12

Multiply $a^{2}$ by $a^{2}$ by adding the exponents.

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Step 1.12.1

Move $a^{2}$ .

$\frac{3 b^{9} \cdot 1}{5 (a^{2} a^{2}) \cdot 3} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.12.2

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

$\frac{3 b^{9} \cdot 1}{5 a^{2 + 2} \cdot 3} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.12.3

Add $2$ and $2$ .

$\frac{3 b^{9} \cdot 1}{5 a^{4} \cdot 3} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.13

Multiply $3 b^{9}$ by $1$ .

$\frac{3 b^{9}}{5 a^{4} \cdot 3} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.14

Cancel the common factor of $3$ .

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Step 1.14.1

Cancel the common factor.

$\frac{3 b^{9}}{5 a^{4} \cdot 3} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.14.2

Rewrite the expression.

$\frac{b^{9}}{5 a^{4}} = \frac{1}{5} \cdot (b^{9} a^{- 4})$

Step 1.15

Remove parentheses.

$\frac{b^{9}}{5 a^{4}} = \frac{1}{5} \cdot b^{9} a^{- 4}$

Step 1.16

Combine $\frac{1}{5}$ and $b^{9}$ .

$\frac{b^{9}}{5 a^{4}} = \frac{b^{9}}{5} a^{- 4}$

Step 1.17

Combine $\frac{b^{9}}{5}$ and $a^{- 4}$ .

$\frac{b^{9}}{5 a^{4}} = \frac{b^{9} a^{- 4}}{5}$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$5 a^{4}, 5$

Step 2.2

Since $5 a^{4}, 5$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $5, 5$ then find LCM for the variable part $a^{4}$ .

Step 2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.4

Since $5$ has no factors besides $1$ and $5$ .

$5$ is a prime number

Step 2.5

The LCM of $5, 5$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$5$

Step 2.6

The factors for $a^{4}$ are $a \cdot a \cdot a \cdot a$ , which is $a$ multiplied by each other $4$ times.

$a^{4} = a \cdot a \cdot a \cdot a$

$a$ occurs $4$ times.

Step 2.7

The LCM of $a^{4}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$a \cdot a \cdot a \cdot a$

Step 2.8

Simplify $a \cdot a \cdot a \cdot a$ .

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Step 2.8.1

Multiply $a$ by $a$ .

$a^{2} \cdot a \cdot a$

Step 2.8.2

Multiply $a^{2}$ by $a$ by adding the exponents.

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Step 2.8.2.1

Multiply $a^{2}$ by $a$ .

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Step 2.8.2.1.1

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

$a^{2} \cdot a^{1} \cdot a$

Step 2.8.2.1.2

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

$a^{2 + 1} \cdot a$

Step 2.8.2.2

Add $2$ and $1$ .

$a^{3} \cdot a$

Step 2.8.3

Multiply $a^{3}$ by $a$ by adding the exponents.

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Step 2.8.3.1

Multiply $a^{3}$ by $a$ .

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Step 2.8.3.1.1

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

$a^{3} \cdot a^{1}$

Step 2.8.3.1.2

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

$a^{3 + 1}$

Step 2.8.3.2

Add $3$ and $1$ .

$a^{4}$

Step 2.9

The LCM for $5 a^{4}, 5$ is the numeric part $5$ multiplied by the variable part.

$5 a^{4}$

Step 3

Multiply each term in $\frac{b^{9}}{5 a^{4}} = \frac{b^{9} a^{- 4}}{5}$ by $5 a^{4}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{b^{9}}{5 a^{4}} = \frac{b^{9} a^{- 4}}{5}$ by $5 a^{4}$ .

$\frac{b^{9}}{5 a^{4}} (5 a^{4}) = \frac{b^{9} a^{- 4}}{5} (5 a^{4})$

Step 3.2

Simplify the left side.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

$5 \frac{b^{9}}{5 a^{4}} a^{4} = \frac{b^{9} a^{- 4}}{5} (5 a^{4})$

Step 3.2.2

Cancel the common factor of $5$ .

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Step 3.2.2.1

Factor $5$ out of $5 a^{4}$ .

$5 \frac{b^{9}}{5 (a^{4})} a^{4} = \frac{b^{9} a^{- 4}}{5} (5 a^{4})$

Step 3.2.2.2

Cancel the common factor.

$5 \frac{b^{9}}{5 a^{4}} a^{4} = \frac{b^{9} a^{- 4}}{5} (5 a^{4})$

Step 3.2.2.3

Rewrite the expression.

$\frac{b^{9}}{a^{4}} a^{4} = \frac{b^{9} a^{- 4}}{5} (5 a^{4})$

Step 3.2.3

Cancel the common factor of $a^{4}$ .

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Step 3.2.3.1

Cancel the common factor.

$\frac{b^{9}}{a^{4}} a^{4} = \frac{b^{9} a^{- 4}}{5} (5 a^{4})$

Step 3.2.3.2

Rewrite the expression.

$b^{9} = \frac{b^{9} a^{- 4}}{5} (5 a^{4})$

Step 3.3

Simplify the right side.

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Step 3.3.1

Rewrite using the commutative property of multiplication.

$b^{9} = 5 \frac{b^{9} a^{- 4}}{5} a^{4}$

Step 3.3.2

Cancel the common factor of $5$ .

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Step 3.3.2.1

Cancel the common factor.

$b^{9} = 5 \frac{b^{9} a^{- 4}}{5} a^{4}$

Step 3.3.2.2

Rewrite the expression.

$b^{9} = b^{9} a^{- 4} a^{4}$

Step 3.3.3

Multiply $a^{- 4}$ by $a^{4}$ by adding the exponents.

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Step 3.3.3.1

Move $a^{4}$ .

$b^{9} = b^{9} (a^{4} a^{- 4})$

Step 3.3.3.2

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

$b^{9} = b^{9} a^{4 - 4}$

Step 3.3.3.3

Subtract $4$ from $4$ .

$b^{9} = b^{9} a^{0}$

Step 3.3.4

Simplify $b^{9} a^{0}$ .

$b^{9} = b^{9}$

Step 4

Solve the equation.

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Step 4.1

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$b = b$

Step 4.2

Solve for $a$ .

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Step 4.2.1

Move all terms containing $b$ to the left side of the equation.

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Step 4.2.1.1

Subtract $b$ from both sides of the equation.

$b - b = 0$

Step 4.2.1.2

Subtract $b$ from $b$ .

$0 = 0$

Step 4.2.2

Since $0 = 0$ , the equation will always be true.

Always true

Step 5

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$