Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $2 y$ .

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

Step 1.2

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

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

Step 2

Multiply $y^{- 1}$ by $y^{- 5}$ by adding the exponents.

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Step 2.1

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

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

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

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