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

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

Step 1

Simplify the numerator.

Apply the product rule to $3 y^{4} n^{6}$ .

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

Apply the product rule to $3 y^{4}$ .

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

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

$\frac{9 {(y^{4})}^{2} {(n^{6})}^{2}}{{(y^{2} n^{- 3})}^{4}}$

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

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

$\frac{9 y^{4 \cdot 2} {(n^{6})}^{2}}{{(y^{2} n^{- 3})}^{4}}$

Multiply $4$ by $2$ .

$\frac{9 y^{8} {(n^{6})}^{2}}{{(y^{2} n^{- 3})}^{4}}$

Multiply the exponents in ${(n^{6})}^{2}$ .

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

$\frac{9 y^{8} n^{6 \cdot 2}}{{(y^{2} n^{- 3})}^{4}}$

Multiply $6$ by $2$ .

$\frac{9 y^{8} n^{12}}{{(y^{2} n^{- 3})}^{4}}$

Step 2

Simplify the denominator.

Apply the product rule to $y^{2} n^{- 3}$ .

$\frac{9 y^{8} n^{12}}{{(y^{2})}^{4} {(n^{- 3})}^{4}}$

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

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

$\frac{9 y^{8} n^{12}}{y^{2 \cdot 4} {(n^{- 3})}^{4}}$

Multiply $2$ by $4$ .

$\frac{9 y^{8} n^{12}}{y^{8} {(n^{- 3})}^{4}}$

Multiply the exponents in ${(n^{- 3})}^{4}$ .

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

$\frac{9 y^{8} n^{12}}{y^{8} n^{- 3 \cdot 4}}$

Multiply $- 3$ by $4$ .

$\frac{9 y^{8} n^{12}}{y^{8} n^{- 12}}$

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

$\frac{9 y^{8} n^{12}}{y^{8} \frac{1}{n^{12}}}$

Step 3

Combine $y^{8}$ and $\frac{1}{n^{12}}$ .

$\frac{9 y^{8} n^{12}}{\frac{y^{8}}{n^{12}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$9 y^{8} n^{12} \frac{n^{12}}{y^{8}}$

Step 5

Cancel the common factor of $y^{8}$ .

Factor $y^{8}$ out of $9 y^{8} n^{12}$ .

$y^{8} (9 n^{12}) \frac{n^{12}}{y^{8}}$

Cancel the common factor.

$y^{8} (9 n^{12}) \frac{n^{12}}{y^{8}}$

Rewrite the expression.

$9 n^{12} n^{12}$

Step 6

Multiply $n^{12}$ by $n^{12}$ by adding the exponents.

Move $n^{12}$ .

$9 (n^{12} n^{12})$

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

$9 n^{12 + 12}$

Add $12$ and $12$ .

$9 n^{24}$

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

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