Simplify the Radical Expression (16x^12y^6)^(-5/2)

${(16 x^{12} y^{6})}^{- \frac{5}{2}}$

Step 1

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

$\frac{1}{{(16 x^{12} y^{6})}^{\frac{5}{2}}}$

Step 2

Simplify the denominator.

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

Apply the product rule to $16 x^{12} y^{6}$ .

$\frac{1}{{(16 x^{12})}^{\frac{5}{2}} {(y^{6})}^{\frac{5}{2}}}$

Apply the product rule to $16 x^{12}$ .

$\frac{1}{16^{\frac{5}{2}} {(x^{12})}^{\frac{5}{2}} {(y^{6})}^{\frac{5}{2}}}$

Rewrite $16$ as $4^{2}$ .

$\frac{1}{{(4^{2})}^{\frac{5}{2}} {(x^{12})}^{\frac{5}{2}} {(y^{6})}^{\frac{5}{2}}}$

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

$\frac{1}{4^{2 (\frac{5}{2})} {(x^{12})}^{\frac{5}{2}} {(y^{6})}^{\frac{5}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{1}{4^{2 (\frac{5}{2})} {(x^{12})}^{\frac{5}{2}} {(y^{6})}^{\frac{5}{2}}}$

Rewrite the expression.

$\frac{1}{4^{5} {(x^{12})}^{\frac{5}{2}} {(y^{6})}^{\frac{5}{2}}}$

Raise $4$ to the power of $5$ .

$\frac{1}{1024 {(x^{12})}^{\frac{5}{2}} {(y^{6})}^{\frac{5}{2}}}$

Multiply the exponents in ${(x^{12})}^{\frac{5}{2}}$ .

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

$\frac{1}{1024 x^{12 (\frac{5}{2})} {(y^{6})}^{\frac{5}{2}}}$

Cancel the common factor of $2$ .

Factor $2$ out of $12$ .

$\frac{1}{1024 x^{2 (6) \frac{5}{2}} {(y^{6})}^{\frac{5}{2}}}$

Cancel the common factor.

$\frac{1}{1024 x^{2 \cdot 6 \frac{5}{2}} {(y^{6})}^{\frac{5}{2}}}$

Rewrite the expression.

$\frac{1}{1024 x^{6 \cdot 5} {(y^{6})}^{\frac{5}{2}}}$

Multiply $6$ by $5$ .

$\frac{1}{1024 x^{30} {(y^{6})}^{\frac{5}{2}}}$

Multiply the exponents in ${(y^{6})}^{\frac{5}{2}}$ .

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

$\frac{1}{1024 x^{30} y^{6 (\frac{5}{2})}}$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$\frac{1}{1024 x^{30} y^{2 (3) \frac{5}{2}}}$

Cancel the common factor.

$\frac{1}{1024 x^{30} y^{2 \cdot 3 \frac{5}{2}}}$

Rewrite the expression.

$\frac{1}{1024 x^{30} y^{3 \cdot 5}}$

Multiply $3$ by $5$ .

$\frac{1}{1024 x^{30} y^{15}}$

Simplify the Radical Expression (16x^12y^6)^(-5/2)

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