Simplify (((2^-3)(x^-3)(y^2))/((4^-2)(x^4)(y^6)))^2

${(\frac{(2^{- 3}) (x^{- 3}) (y^{2})}{(4^{- 2}) (x^{4}) (y^{6})})}^{2}$

Step 1

Move $2^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

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

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

${(\frac{y^{2}}{4^{- 2} x^{3 + 4} y^{6} \cdot 2^{3}})}^{2}$

Add $3$ and $4$ .

${(\frac{y^{2}}{4^{- 2} x^{7} y^{6} \cdot 2^{3}})}^{2}$

Step 4

Move $4^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{y^{2} \cdot 4^{2}}{x^{7} y^{6} \cdot 2^{3}})}^{2}$

Step 5

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

Factor $y^{2}$ out of $y^{2} \cdot 4^{2}$ .

${(\frac{y^{2} (4^{2})}{x^{7} y^{6} \cdot 2^{3}})}^{2}$

Cancel the common factors.

Factor $y^{2}$ out of $x^{7} y^{6} \cdot 2^{3}$ .

${(\frac{y^{2} (4^{2})}{y^{2} (x^{7} y^{4} \cdot 2^{3})})}^{2}$

Cancel the common factor.

${(\frac{y^{2} \cdot 4^{2}}{y^{2} (x^{7} y^{4} \cdot 2^{3})})}^{2}$

Rewrite the expression.

${(\frac{4^{2}}{x^{7} y^{4} \cdot 2^{3}})}^{2}$

Step 6

Evaluate the exponents.

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

${(\frac{16}{x^{7} y^{4} \cdot 2^{3}})}^{2}$

Raise $2$ to the power of $3$ .

${(\frac{16}{x^{7} y^{4} \cdot 8})}^{2}$

Step 7

Cancel the common factor of $16$ and $8$ .

Factor $8$ out of $16$ .

${(\frac{8 \cdot 2}{x^{7} y^{4} \cdot 8})}^{2}$

Cancel the common factors.

Factor $8$ out of $x^{7} y^{4} \cdot 8$ .

${(\frac{8 \cdot 2}{8 \cdot (x^{7} y^{4})})}^{2}$

Cancel the common factor.

${(\frac{8 \cdot 2}{8 \cdot (x^{7} y^{4})})}^{2}$

Rewrite the expression.

${(\frac{2}{x^{7} y^{4}})}^{2}$

Step 8

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

Apply the product rule to $\frac{2}{x^{7} y^{4}}$ .

$\frac{2^{2}}{{(x^{7} y^{4})}^{2}}$

Apply the product rule to $x^{7} y^{4}$ .

$\frac{2^{2}}{{(x^{7})}^{2} {(y^{4})}^{2}}$

Step 9

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

$\frac{4}{{(x^{7})}^{2} {(y^{4})}^{2}}$

Step 10

Simplify the denominator.

Multiply the exponents in ${(x^{7})}^{2}$ .

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

$\frac{4}{x^{7 \cdot 2} {(y^{4})}^{2}}$

Multiply $7$ by $2$ .

$\frac{4}{x^{14} {(y^{4})}^{2}}$

Multiply the exponents in ${(y^{4})}^{2}$ .

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

$\frac{4}{x^{14} y^{4 \cdot 2}}$

Multiply $4$ by $2$ .

$\frac{4}{x^{14} y^{8}}$

Simplify (((2^-3)(x^-3)(y^2))/((4^-2)(x^4)(y^6)))^2

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