Simplify ((16x^6y^8)^(1/2))/((81x^12y^4)^(3/4))

$\frac{{(16 x^{6} y^{8})}^{\frac{1}{2}}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

Step 1

Simplify the numerator.

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

Apply the product rule to $16 x^{6} y^{8}$ .

$\frac{{(16 x^{6})}^{\frac{1}{2}} {(y^{8})}^{\frac{1}{2}}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

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

$\frac{16^{\frac{1}{2}} {(x^{6})}^{\frac{1}{2}} {(y^{8})}^{\frac{1}{2}}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

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

$\frac{{(4^{2})}^{\frac{1}{2}} {(x^{6})}^{\frac{1}{2}} {(y^{8})}^{\frac{1}{2}}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

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

$\frac{4^{2 (\frac{1}{2})} {(x^{6})}^{\frac{1}{2}} {(y^{8})}^{\frac{1}{2}}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4^{2 (\frac{1}{2})} {(x^{6})}^{\frac{1}{2}} {(y^{8})}^{\frac{1}{2}}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

Rewrite the expression.

$\frac{4^{1} {(x^{6})}^{\frac{1}{2}} {(y^{8})}^{\frac{1}{2}}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

Evaluate the exponent.

$\frac{4 {(x^{6})}^{\frac{1}{2}} {(y^{8})}^{\frac{1}{2}}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

Multiply the exponents in ${(x^{6})}^{\frac{1}{2}}$ .

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

$\frac{4 x^{6 (\frac{1}{2})} {(y^{8})}^{\frac{1}{2}}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$\frac{4 x^{2 (3) \frac{1}{2}} {(y^{8})}^{\frac{1}{2}}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

Cancel the common factor.

$\frac{4 x^{2 \cdot 3 \frac{1}{2}} {(y^{8})}^{\frac{1}{2}}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

Rewrite the expression.

$\frac{4 x^{3} {(y^{8})}^{\frac{1}{2}}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

Multiply the exponents in ${(y^{8})}^{\frac{1}{2}}$ .

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

$\frac{4 x^{3} y^{8 (\frac{1}{2})}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

$\frac{4 x^{3} y^{2 (4) \frac{1}{2}}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

Cancel the common factor.

$\frac{4 x^{3} y^{2 \cdot 4 \frac{1}{2}}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

Rewrite the expression.

$\frac{4 x^{3} y^{4}}{{(81 x^{12} y^{4})}^{\frac{3}{4}}}$

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 $81 x^{12} y^{4}$ .

$\frac{4 x^{3} y^{4}}{{(81 x^{12})}^{\frac{3}{4}} {(y^{4})}^{\frac{3}{4}}}$

Apply the product rule to $81 x^{12}$ .

$\frac{4 x^{3} y^{4}}{81^{\frac{3}{4}} {(x^{12})}^{\frac{3}{4}} {(y^{4})}^{\frac{3}{4}}}$

Rewrite $81$ as $3^{4}$ .

$\frac{4 x^{3} y^{4}}{{(3^{4})}^{\frac{3}{4}} {(x^{12})}^{\frac{3}{4}} {(y^{4})}^{\frac{3}{4}}}$

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

$\frac{4 x^{3} y^{4}}{3^{4 (\frac{3}{4})} {(x^{12})}^{\frac{3}{4}} {(y^{4})}^{\frac{3}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{3} y^{4}}{3^{4 (\frac{3}{4})} {(x^{12})}^{\frac{3}{4}} {(y^{4})}^{\frac{3}{4}}}$

Rewrite the expression.

$\frac{4 x^{3} y^{4}}{3^{3} {(x^{12})}^{\frac{3}{4}} {(y^{4})}^{\frac{3}{4}}}$

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

$\frac{4 x^{3} y^{4}}{27 {(x^{12})}^{\frac{3}{4}} {(y^{4})}^{\frac{3}{4}}}$

Multiply the exponents in ${(x^{12})}^{\frac{3}{4}}$ .

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

$\frac{4 x^{3} y^{4}}{27 x^{12 (\frac{3}{4})} {(y^{4})}^{\frac{3}{4}}}$

Cancel the common factor of $4$ .

Factor $4$ out of $12$ .

$\frac{4 x^{3} y^{4}}{27 x^{4 (3) \frac{3}{4}} {(y^{4})}^{\frac{3}{4}}}$

Cancel the common factor.

$\frac{4 x^{3} y^{4}}{27 x^{4 \cdot 3 \frac{3}{4}} {(y^{4})}^{\frac{3}{4}}}$

Rewrite the expression.

$\frac{4 x^{3} y^{4}}{27 x^{3 \cdot 3} {(y^{4})}^{\frac{3}{4}}}$

Multiply $3$ by $3$ .

$\frac{4 x^{3} y^{4}}{27 x^{9} {(y^{4})}^{\frac{3}{4}}}$

Multiply the exponents in ${(y^{4})}^{\frac{3}{4}}$ .

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

$\frac{4 x^{3} y^{4}}{27 x^{9} y^{4 (\frac{3}{4})}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{3} y^{4}}{27 x^{9} y^{4 (\frac{3}{4})}}$

Rewrite the expression.

$\frac{4 x^{3} y^{4}}{27 x^{9} y^{3}}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x^{3}$ and $x^{9}$ .

Factor $x^{3}$ out of $4 x^{3} y^{4}$ .

$\frac{x^{3} (4 y^{4})}{27 x^{9} y^{3}}$

Cancel the common factors.

Factor $x^{3}$ out of $27 x^{9} y^{3}$ .

$\frac{x^{3} (4 y^{4})}{x^{3} (27 x^{6} y^{3})}$

Cancel the common factor.

$\frac{x^{3} (4 y^{4})}{x^{3} (27 x^{6} y^{3})}$

Rewrite the expression.

$\frac{4 y^{4}}{27 x^{6} y^{3}}$

Cancel the common factor of $y^{4}$ and $y^{3}$ .

Factor $y^{3}$ out of $4 y^{4}$ .

$\frac{y^{3} (4 y)}{27 x^{6} y^{3}}$

Cancel the common factors.

Factor $y^{3}$ out of $27 x^{6} y^{3}$ .

$\frac{y^{3} (4 y)}{y^{3} (27 x^{6})}$

Cancel the common factor.

$\frac{y^{3} (4 y)}{y^{3} (27 x^{6})}$

Rewrite the expression.

$\frac{4 y}{27 x^{6}}$

Simplify ((16x^6y^8)^(1/2))/((81x^12y^4)^(3/4))

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