Calculus Examples

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

Step 1

Move ${(y^{2} x^{- 4})}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Apply the product rule to $y^{2} x^{- 4}$ .

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

Move ${(y^{2})}^{2}$ .

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

Step 2.3.2

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

$\frac{y^{9}}{3^{4} {(y^{2})}^{2 + 4} x^{2} {(x^{- 4})}^{2}}$

Step 2.3.3

Add $2$ and $4$ .

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

Step 2.4

Raise $3$ to the power of $4$ .

$\frac{y^{9}}{81 {(y^{2})}^{6} x^{2} {(x^{- 4})}^{2}}$

Step 2.5

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

Tap for more steps...

Step 2.5.1

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

$\frac{y^{9}}{81 y^{2 \cdot 6} x^{2} {(x^{- 4})}^{2}}$

Step 2.5.2

Multiply $2$ by $6$ .

$\frac{y^{9}}{81 y^{12} x^{2} {(x^{- 4})}^{2}}$

Step 2.6

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

Tap for more steps...

Step 2.6.1

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

$\frac{y^{9}}{81 y^{12} x^{2} x^{- 4 \cdot 2}}$

Step 2.6.2

Multiply $- 4$ by $2$ .

$\frac{y^{9}}{81 y^{12} x^{2} x^{- 8}}$

Step 2.7

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

$\frac{y^{9}}{81 y^{12} x^{2} \frac{1}{x^{8}}}$

Step 2.8

Combine exponents.

Tap for more steps...

Step 2.8.1

Combine $81$ and $\frac{1}{x^{8}}$ .

$\frac{y^{9}}{y^{12} x^{2} \frac{81}{x^{8}}}$

Step 2.8.2

Combine $y^{12}$ and $\frac{81}{x^{8}}$ .

$\frac{y^{9}}{x^{2} \frac{y^{12} \cdot 81}{x^{8}}}$

Step 2.8.3

Combine $x^{2}$ and $\frac{y^{12} \cdot 81}{x^{8}}$ .

$\frac{y^{9}}{\frac{x^{2} (y^{12} \cdot 81)}{x^{8}}}$

Step 2.9

Remove unnecessary parentheses.

$\frac{y^{9}}{\frac{x^{2} y^{12} \cdot 81}{x^{8}}}$

Step 2.10

Reduce the expression $\frac{x^{2} y^{12} \cdot 81}{x^{8}}$ by cancelling the common factors.

Tap for more steps...

Step 2.10.1

Factor $x^{2}$ out of $x^{2} y^{12} \cdot 81$ .

$\frac{y^{9}}{\frac{x^{2} (y^{12} \cdot 81)}{x^{8}}}$

Step 2.10.2

Factor $x^{2}$ out of $x^{8}$ .

$\frac{y^{9}}{\frac{x^{2} (y^{12} \cdot 81)}{x^{2} x^{6}}}$

Step 2.10.3

Cancel the common factor.

$\frac{y^{9}}{\frac{x^{2} (y^{12} \cdot 81)}{x^{2} x^{6}}}$

Step 2.10.4

Rewrite the expression.

$\frac{y^{9}}{\frac{y^{12} \cdot 81}{x^{6}}}$

Step 2.11

Move $81$ to the left of $y^{12}$ .

$\frac{y^{9}}{\frac{81 y^{12}}{x^{6}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$y^{9} \frac{x^{6}}{81 y^{12}}$

Step 4

Cancel the common factor of $y^{9}$ .

Tap for more steps...

Step 4.1

Factor $y^{9}$ out of $81 y^{12}$ .

$y^{9} \frac{x^{6}}{y^{9} (81 y^{3})}$

Step 4.2

Cancel the common factor.

$y^{9} \frac{x^{6}}{y^{9} (81 y^{3})}$

Step 4.3

Rewrite the expression.

$\frac{x^{6}}{81 y^{3}}$