Write the Fraction in Simplest Form ((4z^-2x^3)^2)/(2z^5x^2)

$\frac{{(4 z^{- 2} x^{3})}^{2}}{2 z^{5} x^{2}}$

Step 1

Simplify the numerator.

Apply the product rule to $4 z^{- 2} x^{3}$ .

$\frac{{(4 z^{- 2})}^{2} {(x^{3})}^{2}}{2 z^{5} x^{2}}$

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

$\frac{{(4 \frac{1}{z^{2}})}^{2} {(x^{3})}^{2}}{2 z^{5} x^{2}}$

Combine $4$ and $\frac{1}{z^{2}}$ .

$\frac{{(\frac{4}{z^{2}})}^{2} {(x^{3})}^{2}}{2 z^{5} x^{2}}$

Apply the product rule to $\frac{4}{z^{2}}$ .

$\frac{\frac{4^{2}}{{(z^{2})}^{2}} {(x^{3})}^{2}}{2 z^{5} x^{2}}$

Raise $4$ to the power of $2$ .

$\frac{\frac{16}{{(z^{2})}^{2}} {(x^{3})}^{2}}{2 z^{5} x^{2}}$

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

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

$\frac{\frac{16}{z^{2 \cdot 2}} {(x^{3})}^{2}}{2 z^{5} x^{2}}$

Multiply $2$ by $2$ .

$\frac{\frac{16}{z^{4}} {(x^{3})}^{2}}{2 z^{5} x^{2}}$

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

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

$\frac{\frac{16}{z^{4}} x^{3 \cdot 2}}{2 z^{5} x^{2}}$

Multiply $3$ by $2$ .

$\frac{\frac{16}{z^{4}} x^{6}}{2 z^{5} x^{2}}$

Step 2

Combine $\frac{16}{z^{4}}$ and $x^{6}$ .

$\frac{\frac{16 x^{6}}{z^{4}}}{2 z^{5} x^{2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{16 x^{6}}{z^{4}} \cdot \frac{1}{2 z^{5} x^{2}}$

Step 4

Combine.

$\frac{16 x^{6} \cdot 1}{z^{4} (2 z^{5} x^{2})}$

Step 5

Multiply $z^{4}$ by $z^{5}$ by adding the exponents.

Move $z^{5}$ .

$\frac{16 x^{6} \cdot 1}{z^{5} z^{4} (2 x^{2})}$

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

$\frac{16 x^{6} \cdot 1}{z^{5 + 4} (2 x^{2})}$

Add $5$ and $4$ .

$\frac{16 x^{6} \cdot 1}{z^{9} (2 x^{2})}$

Step 6

Multiply $16 x^{6}$ by $1$ .

$\frac{16 x^{6}}{z^{9} (2 x^{2})}$

Step 7

Cancel the common factor of $16$ and $2$ .

Factor $2$ out of $16 x^{6}$ .

$\frac{2 (8 x^{6})}{z^{9} (2 x^{2})}$

Cancel the common factors.

Factor $2$ out of $z^{9} (2 x^{2})$ .

$\frac{2 (8 x^{6})}{2 (z^{9} (x^{2}))}$

Cancel the common factor.

$\frac{2 (8 x^{6})}{2 (z^{9} (x^{2}))}$

Rewrite the expression.

$\frac{8 x^{6}}{z^{9} (x^{2})}$

Step 8

Cancel the common factor of $x^{6}$ and $x^{2}$ .

Factor $x^{2}$ out of $8 x^{6}$ .

$\frac{x^{2} (8 x^{4})}{z^{9} (x^{2})}$

Cancel the common factors.

Factor $x^{2}$ out of $z^{9} (x^{2})$ .

$\frac{x^{2} (8 x^{4})}{x^{2} z^{9}}$

Cancel the common factor.

$\frac{x^{2} (8 x^{4})}{x^{2} z^{9}}$

Rewrite the expression.

$\frac{8 x^{4}}{z^{9}}$

Write the Fraction in Simplest Form ((4z^-2x^3)^2)/(2z^5x^2)

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