Simplify 1/(4x^5)*((2x^2)/(y^3))^5

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

Step 1

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

Apply the product rule to $\frac{2 x^{2}}{y^{3}}$ .

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

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

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

Step 2

Combine fractions.

Combine.

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

Simplify the expression.

Multiply $2^{5}$ by $1$ .

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

Multiply the exponents in ${(y^{3})}^{5}$ .

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

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

Multiply $3$ by $5$ .

$\frac{2^{5} {(x^{2})}^{5}}{4 x^{5} y^{15}}$

Step 3

Simplify the numerator.

Raise $2$ to the power of $5$ .

$\frac{32 {(x^{2})}^{5}}{4 x^{5} y^{15}}$

Multiply the exponents in ${(x^{2})}^{5}$ .

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

$\frac{32 x^{2 \cdot 5}}{4 x^{5} y^{15}}$

Multiply $2$ by $5$ .

$\frac{32 x^{10}}{4 x^{5} y^{15}}$

Step 4

Reduce the expression by cancelling the common factors.

Cancel the common factor of $32$ and $4$ .

Factor $4$ out of $32 x^{10}$ .

$\frac{4 (8 x^{10})}{4 x^{5} y^{15}}$

Cancel the common factors.

Factor $4$ out of $4 x^{5} y^{15}$ .

$\frac{4 (8 x^{10})}{4 (x^{5} y^{15})}$

Cancel the common factor.

$\frac{4 (8 x^{10})}{4 (x^{5} y^{15})}$

Rewrite the expression.

$\frac{8 x^{10}}{x^{5} y^{15}}$

Cancel the common factor of $x^{10}$ and $x^{5}$ .

Factor $x^{5}$ out of $8 x^{10}$ .

$\frac{x^{5} (8 x^{5})}{x^{5} y^{15}}$

Cancel the common factors.

Factor $x^{5}$ out of $x^{5} y^{15}$ .

$\frac{x^{5} (8 x^{5})}{x^{5} (y^{15})}$

Cancel the common factor.

$\frac{x^{5} (8 x^{5})}{x^{5} y^{15}}$

Rewrite the expression.

$\frac{8 x^{5}}{y^{15}}$

Simplify 1/(4x^5)*((2x^2)/(y^3))^5

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