Basic Math Examples

${(\frac{5 z^{3}}{2 a^{2}})}^{- 3} {(\frac{8 a^{- 1}}{15 z^{- 2}})}^{- 3}$

Step 1

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{2 a^{2}}{5 z^{3}})}^{3} {(\frac{8 a^{- 1}}{15 z^{- 2}})}^{- 3}$

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1

Apply the product rule to $\frac{2 a^{2}}{5 z^{3}}$ .

$\frac{{(2 a^{2})}^{3}}{{(5 z^{3})}^{3}} {(\frac{8 a^{- 1}}{15 z^{- 2}})}^{- 3}$

Step 2.2

Apply the product rule to $2 a^{2}$ .

$\frac{2^{3} {(a^{2})}^{3}}{{(5 z^{3})}^{3}} {(\frac{8 a^{- 1}}{15 z^{- 2}})}^{- 3}$

Step 2.3

Apply the product rule to $5 z^{3}$ .

$\frac{2^{3} {(a^{2})}^{3}}{5^{3} {(z^{3})}^{3}} {(\frac{8 a^{- 1}}{15 z^{- 2}})}^{- 3}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $2$ to the power of $3$ .

$\frac{8 {(a^{2})}^{3}}{5^{3} {(z^{3})}^{3}} {(\frac{8 a^{- 1}}{15 z^{- 2}})}^{- 3}$

Step 3.2

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

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Step 3.2.1

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

$\frac{8 a^{2 \cdot 3}}{5^{3} {(z^{3})}^{3}} {(\frac{8 a^{- 1}}{15 z^{- 2}})}^{- 3}$

Step 3.2.2

Multiply $2$ by $3$ .

$\frac{8 a^{6}}{5^{3} {(z^{3})}^{3}} {(\frac{8 a^{- 1}}{15 z^{- 2}})}^{- 3}$

Step 4

Simplify the denominator.

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Step 4.1

Raise $5$ to the power of $3$ .

$\frac{8 a^{6}}{125 {(z^{3})}^{3}} {(\frac{8 a^{- 1}}{15 z^{- 2}})}^{- 3}$

Step 4.2

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

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Step 4.2.1

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

$\frac{8 a^{6}}{125 z^{3 \cdot 3}} {(\frac{8 a^{- 1}}{15 z^{- 2}})}^{- 3}$

Step 4.2.2

Multiply $3$ by $3$ .

$\frac{8 a^{6}}{125 z^{9}} {(\frac{8 a^{- 1}}{15 z^{- 2}})}^{- 3}$

Step 5

Simplify the expression.

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Step 5.1

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

$\frac{8 a^{6}}{125 z^{9}} {(\frac{8}{15 z^{- 2} a})}^{- 3}$

Step 5.2

Move $z^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{8 a^{6}}{125 z^{9}} {(\frac{8 z^{2}}{15 a})}^{- 3}$

Step 6

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{8 a^{6}}{125 z^{9}} {(\frac{15 a}{8 z^{2}})}^{3}$

Step 7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.1

Apply the product rule to $\frac{15 a}{8 z^{2}}$ .

$\frac{8 a^{6}}{125 z^{9}} \cdot \frac{{(15 a)}^{3}}{{(8 z^{2})}^{3}}$

Step 7.2

Apply the product rule to $15 a$ .

$\frac{8 a^{6}}{125 z^{9}} \cdot \frac{15^{3} a^{3}}{{(8 z^{2})}^{3}}$

Step 7.3

Apply the product rule to $8 z^{2}$ .

$\frac{8 a^{6}}{125 z^{9}} \cdot \frac{15^{3} a^{3}}{8^{3} {(z^{2})}^{3}}$

Step 8

Combine.

$\frac{8 a^{6} (15^{3} a^{3})}{125 z^{9} (8^{3} {(z^{2})}^{3})}$

Step 9

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

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Step 9.1

Move $a^{3}$ .

$\frac{8 (a^{3} a^{6}) \cdot 15^{3}}{125 z^{9} (8^{3} {(z^{2})}^{3})}$

Step 9.2

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

$\frac{8 a^{3 + 6} \cdot 15^{3}}{125 z^{9} (8^{3} {(z^{2})}^{3})}$

Step 9.3

Add $3$ and $6$ .

$\frac{8 a^{9} \cdot 15^{3}}{125 z^{9} (8^{3} {(z^{2})}^{3})}$

Step 10

Reduce the expression by cancelling the common factors.

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Step 10.1

Cancel the common factor of $8$ and $8^{3}$ .

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Step 10.1.1

Factor $8$ out of $8 a^{9} \cdot 15^{3}$ .

$\frac{8 (a^{9} \cdot 15^{3})}{125 z^{9} (8^{3} {(z^{2})}^{3})}$

Step 10.1.2

Cancel the common factors.

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Step 10.1.2.1

Factor $8$ out of $125 z^{9} (8^{3} {(z^{2})}^{3})$ .

$\frac{8 (a^{9} \cdot 15^{3})}{8 (125 z^{9} (8^{2} {(z^{2})}^{3}))}$

Step 10.1.2.2

Cancel the common factor.

$\frac{8 (a^{9} \cdot 15^{3})}{8 (125 z^{9} (8^{2} {(z^{2})}^{3}))}$

Step 10.1.2.3

Rewrite the expression.

$\frac{a^{9} \cdot 15^{3}}{125 z^{9} (8^{2} {(z^{2})}^{3})}$

Step 10.2

Raise $15$ to the power of $3$ .

$\frac{a^{9} \cdot 3375}{125 z^{9} (8^{2} {(z^{2})}^{3})}$

Step 11

Simplify the denominator.

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Step 11.1

Rewrite.

$\frac{a^{9} \cdot 3375}{125 z^{9} (8^{3} {(z^{2})}^{3})}$

Step 11.2

Factor $8$ out of $8^{3}$ .

$\frac{a^{9} \cdot 3375}{125 z^{9} (8 \cdot 8^{2} {(z^{2})}^{3})}$

Step 11.3

Rewrite.

$\frac{a^{9} \cdot 3375}{125 z^{9} (8^{2} {(z^{2})}^{3})}$

Step 11.4

Remove unnecessary parentheses.

$\frac{a^{9} \cdot 3375}{125 z^{9} \cdot 8^{2} {(z^{2})}^{3}}$

Step 11.5

Raise $8$ to the power of $2$ .

$\frac{a^{9} \cdot 3375}{125 z^{9} \cdot 64 {(z^{2})}^{3}}$

Step 11.6

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

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Step 11.6.1

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

$\frac{a^{9} \cdot 3375}{125 z^{9} \cdot 64 z^{2 \cdot 3}}$

Step 11.6.2

Multiply $2$ by $3$ .

$\frac{a^{9} \cdot 3375}{125 z^{9} \cdot 64 z^{6}}$

Step 11.7

Combine exponents.

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Step 11.7.1

Multiply $64$ by $125$ .

$\frac{a^{9} \cdot 3375}{8000 z^{9} z^{6}}$

Step 11.7.2

Multiply $z^{9}$ by $z^{6}$ by adding the exponents.

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Step 11.7.2.1

Move $z^{6}$ .

$\frac{a^{9} \cdot 3375}{8000 (z^{6} z^{9})}$

Step 11.7.2.2

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

$\frac{a^{9} \cdot 3375}{8000 z^{6 + 9}}$

Step 11.7.2.3

Add $6$ and $9$ .

$\frac{a^{9} \cdot 3375}{8000 z^{15}}$

Step 12

Reduce the expression by cancelling the common factors.

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Step 12.1

Cancel the common factor of $3375$ and $8000$ .

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Step 12.1.1

Factor $125$ out of $a^{9} \cdot 3375$ .

$\frac{125 (a^{9} \cdot 27)}{8000 z^{15}}$

Step 12.1.2

Cancel the common factors.

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Step 12.1.2.1

Factor $125$ out of $8000 z^{15}$ .

$\frac{125 (a^{9} \cdot 27)}{125 (64 z^{15})}$

Step 12.1.2.2

Cancel the common factor.

$\frac{125 (a^{9} \cdot 27)}{125 (64 z^{15})}$

Step 12.1.2.3

Rewrite the expression.

$\frac{a^{9} \cdot 27}{64 z^{15}}$

Step 12.2

Move $27$ to the left of $a^{9}$ .

$\frac{27 a^{9}}{64 z^{15}}$