Algebra Examples

${(y^{- 2})}^{4} + \frac{y^{3}}{y^{5}}$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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

$y^{- 2 \cdot 4} + \frac{y^{3}}{y^{5}}$

Step 1.1.2

Multiply $- 2$ by $4$ .

$y^{- 8} + \frac{y^{3}}{y^{5}}$

Step 1.2

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

$\frac{1}{y^{8}} + \frac{y^{3}}{y^{5}}$

Step 1.3

Cancel the common factor of $y^{3}$ and $y^{5}$ .

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Step 1.3.1

Multiply by $1$ .

$\frac{1}{y^{8}} + \frac{y^{3} \cdot 1}{y^{5}}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $y^{3}$ out of $y^{5}$ .

$\frac{1}{y^{8}} + \frac{y^{3} \cdot 1}{y^{3} y^{2}}$

Step 1.3.2.2

Cancel the common factor.

$\frac{1}{y^{8}} + \frac{y^{3} \cdot 1}{y^{3} y^{2}}$

Step 1.3.2.3

Rewrite the expression.

$\frac{1}{y^{8}} + \frac{1}{y^{2}}$

Step 2

To write $\frac{1}{y^{2}}$ as a fraction with a common denominator, multiply by $\frac{y^{6}}{y^{6}}$ .

$\frac{1}{y^{8}} + \frac{1}{y^{2}} \cdot \frac{y^{6}}{y^{6}}$

Step 3

Write each expression with a common denominator of $y^{8}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{y^{2}}$ by $\frac{y^{6}}{y^{6}}$ .

$\frac{1}{y^{8}} + \frac{y^{6}}{y^{2} y^{6}}$

Step 3.2

Multiply $y^{2}$ by $y^{6}$ by adding the exponents.

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Step 3.2.1

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

$\frac{1}{y^{8}} + \frac{y^{6}}{y^{2 + 6}}$

Step 3.2.2

Add $2$ and $6$ .

$\frac{1}{y^{8}} + \frac{y^{6}}{y^{8}}$

Step 4

Combine the numerators over the common denominator.

$\frac{1 + y^{6}}{y^{8}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $1$ as $1^{3}$ .

$\frac{1^{3} + y^{6}}{y^{8}}$

Step 5.2

Rewrite $y^{6}$ as ${(y^{2})}^{3}$ .

$\frac{1^{3} + {(y^{2})}^{3}}{y^{8}}$

Step 5.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 1$ and $b = y^{2}$ .

$\frac{(1 + y^{2}) (1^{2} - 1 y^{2} + {(y^{2})}^{2})}{y^{8}}$

Step 5.4

Simplify.

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Step 5.4.1

One to any power is one.

$\frac{(1 + y^{2}) (1 - 1 y^{2} + {(y^{2})}^{2})}{y^{8}}$

Step 5.4.2

Rewrite $- 1 y^{2}$ as $- y^{2}$ .

$\frac{(1 + y^{2}) (1 - y^{2} + {(y^{2})}^{2})}{y^{8}}$

Step 5.4.3

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

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Step 5.4.3.1

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

$\frac{(1 + y^{2}) (1 - y^{2} + y^{2 \cdot 2})}{y^{8}}$

Step 5.4.3.2

Multiply $2$ by $2$ .

$\frac{(1 + y^{2}) (1 - y^{2} + y^{4})}{y^{8}}$