Write the Fraction in Simplest Form (20z^10)/((5z^-2)^3)

$\frac{20 z^{10}}{{(5 z^{- 2})}^{3}}$

Step 1

Simplify the denominator.

Apply the product rule to $5 z^{- 2}$ .

$\frac{20 z^{10}}{5^{3} {(z^{- 2})}^{3}}$

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

$\frac{20 z^{10}}{125 {(z^{- 2})}^{3}}$

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

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

$\frac{20 z^{10}}{125 z^{- 2 \cdot 3}}$

Multiply $- 2$ by $3$ .

$\frac{20 z^{10}}{125 z^{- 6}}$

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

$\frac{20 z^{10}}{125 \frac{1}{z^{6}}}$

Step 2

Combine $125$ and $\frac{1}{z^{6}}$ .

$\frac{20 z^{10}}{\frac{125}{z^{6}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$20 z^{10} \frac{z^{6}}{125}$

Step 4

Cancel the common factor of $5$ .

Factor $5$ out of $20 z^{10}$ .

$5 (4 z^{10}) \frac{z^{6}}{125}$

Factor $5$ out of $125$ .

$5 (4 z^{10}) \frac{z^{6}}{5 (25)}$

Cancel the common factor.

$5 (4 z^{10}) \frac{z^{6}}{5 \cdot 25}$

Rewrite the expression.

$4 z^{10} \frac{z^{6}}{25}$

Step 5

Combine $4$ and $\frac{z^{6}}{25}$ .

$z^{10} \frac{4 z^{6}}{25}$

Step 6

Combine $z^{10}$ and $\frac{4 z^{6}}{25}$ .

$\frac{z^{10} (4 z^{6})}{25}$

Step 7

Multiply $z^{10}$ by $z^{6}$ by adding the exponents.

Move $z^{6}$ .

$\frac{z^{6} z^{10} \cdot 4}{25}$

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

$\frac{z^{6 + 10} \cdot 4}{25}$

Add $6$ and $10$ .

$\frac{z^{16} \cdot 4}{25}$

Step 8

Move $4$ to the left of $z^{16}$ .

$\frac{4 z^{16}}{25}$

Write the Fraction in Simplest Form (20z^10)/((5z^-2)^3)

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