Basic Math Examples

\frac{{(2 y)}^{2} z^{3}}{y^{- 1} y^{- 5}} \cdot {(5 y^{- 2})}^{- 3}

$\frac{{(2 y)}^{2} z^{3}}{y^{- 1} y^{- 5}} \cdot {(5 y^{- 2})}^{- 3}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to

2 y

$2 y$ .

\frac{2^{2} y^{2} z^{3}}{y^{- 1} y^{- 5}} \cdot {(5 y^{- 2})}^{- 3}

$\frac{2^{2} y^{2} z^{3}}{y^{- 1} y^{- 5}} \cdot {(5 y^{- 2})}^{- 3}$

Step 1.2

Raise

2

$2$ to the power of

2

$2$ .

\frac{4 y^{2} z^{3}}{y^{- 1} y^{- 5}} \cdot {(5 y^{- 2})}^{- 3}

$\frac{4 y^{2} z^{3}}{y^{- 1} y^{- 5}} \cdot {(5 y^{- 2})}^{- 3}$

\frac{4 y^{2} z^{3}}{y^{- 1} y^{- 5}} \cdot {(5 y^{- 2})}^{- 3}

$\frac{4 y^{2} z^{3}}{y^{- 1} y^{- 5}} \cdot {(5 y^{- 2})}^{- 3}$

Step 2

Multiply

y^{- 1}

$y^{- 1}$ by

y^{- 5}

$y^{- 5}$ by adding the exponents.

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Step 2.1

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{4 y^{2} z^{3}}{y^{- 1 - 5}} \cdot {(5 y^{- 2})}^{- 3}

$\frac{4 y^{2} z^{3}}{y^{- 1 - 5}} \cdot {(5 y^{- 2})}^{- 3}$

Step 2.2

Subtract

$5$ from

$- 1$ .

$\frac{4 y^{2} z^{3}}{y^{- 6}} \cdot {(5 y^{- 2})}^{- 3}$

Step 3

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4 y^{2} z^{3}}{\frac{1}{y^{6}}} \cdot {(5 y^{- 2})}^{- 3}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$4 y^{2} z^{3} y^{6} \cdot {(5 y^{- 2})}^{- 3}$

Step 5

Multiply

$y^{2}$ by

$y^{6}$ by adding the exponents.

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Step 5.1

Move

$y^{6}$ .

$4 (y^{6} y^{2}) z^{3} \cdot {(5 y^{- 2})}^{- 3}$

Step 5.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 y^{6 + 2} z^{3} \cdot {(5 y^{- 2})}^{- 3}$

Step 5.3

Add

$6$ and

$2$ .

$4 y^{8} z^{3} \cdot {(5 y^{- 2})}^{- 3}$

Step 6

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$4 y^{8} z^{3} \cdot {(5 \frac{1}{y^{2}})}^{- 3}$

Step 7

Combine

$5$ and

$\frac{1}{y^{2}}$ .

$4 y^{8} z^{3} \cdot {(\frac{5}{y^{2}})}^{- 3}$

Step 8

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

$4 y^{8} z^{3} \cdot {(\frac{y^{2}}{5})}^{3}$

Step 9

Apply the product rule to

$\frac{y^{2}}{5}$ .

$4 y^{8} z^{3} \cdot \frac{{(y^{2})}^{3}}{5^{3}}$

Step 10

Multiply the exponents in

${(y^{2})}^{3}$ .

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Step 10.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$4 y^{8} z^{3} \cdot \frac{y^{2 \cdot 3}}{5^{3}}$

Step 10.2

Multiply

$2$ by

$3$ .

$4 y^{8} z^{3} \cdot \frac{y^{6}}{5^{3}}$

Step 11

Raise

$5$ to the power of

$3$ .

$4 y^{8} z^{3} \cdot \frac{y^{6}}{125}$

Step 12

Multiply

$4 y^{8} z^{3} \frac{y^{6}}{125}$ .

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Step 12.1

Combine

$4$ and

$\frac{y^{6}}{125}$ .

$y^{8} z^{3} \frac{4 y^{6}}{125}$

Step 12.2

Combine

$y^{8}$ and

$\frac{4 y^{6}}{125}$ .

$z^{3} \frac{y^{8} (4 y^{6})}{125}$

Step 12.3

Multiply

$y^{8}$ by

$y^{6}$ by adding the exponents.

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Step 12.3.1

Move

$y^{6}$ .

$z^{3} \frac{y^{6} y^{8} \cdot 4}{125}$

Step 12.3.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$z^{3} \frac{y^{6 + 8} \cdot 4}{125}$

Step 12.3.3

Add

$6$ and

$8$ .

$z^{3} \frac{y^{14} \cdot 4}{125}$

Step 12.4

Combine

$z^{3}$ and

$\frac{y^{14} \cdot 4}{125}$ .

$\frac{z^{3} (y^{14} \cdot 4)}{125}$

Step 13

Move

$4$ to the left of

$z^{3} y^{14}$ .

$\frac{4 z^{3} y^{14}}{125}$