Basic Math Examples

${(\frac{a^{- 5} y^{4}}{a^{2} y^{- 2}})}^{2} {(\frac{a^{4} y^{- 5}}{a^{3} y^{- 7}})}^{- 3}$

Step 1

Move $a^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{y^{4}}{a^{2} y^{- 2} a^{5}})}^{2} {(\frac{a^{4} y^{- 5}}{a^{3} y^{- 7}})}^{- 3}$

Step 2

Multiply $a^{2}$ by $a^{5}$ by adding the exponents.

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Step 2.1

Move $a^{5}$ .

${(\frac{y^{4}}{a^{5} a^{2} y^{- 2}})}^{2} {(\frac{a^{4} y^{- 5}}{a^{3} y^{- 7}})}^{- 3}$

Step 2.2

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

${(\frac{y^{4}}{a^{5 + 2} y^{- 2}})}^{2} {(\frac{a^{4} y^{- 5}}{a^{3} y^{- 7}})}^{- 3}$

Step 2.3

Add $5$ and $2$ .

${(\frac{y^{4}}{a^{7} y^{- 2}})}^{2} {(\frac{a^{4} y^{- 5}}{a^{3} y^{- 7}})}^{- 3}$

Step 3

Move $y^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{y^{4} y^{2}}{a^{7}})}^{2} {(\frac{a^{4} y^{- 5}}{a^{3} y^{- 7}})}^{- 3}$

Step 4

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

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Step 4.1

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

${(\frac{y^{4 + 2}}{a^{7}})}^{2} {(\frac{a^{4} y^{- 5}}{a^{3} y^{- 7}})}^{- 3}$

Step 4.2

Add $4$ and $2$ .

${(\frac{y^{6}}{a^{7}})}^{2} {(\frac{a^{4} y^{- 5}}{a^{3} y^{- 7}})}^{- 3}$

Step 5

Apply the product rule to $\frac{y^{6}}{a^{7}}$ .

$\frac{{(y^{6})}^{2}}{{(a^{7})}^{2}} {(\frac{a^{4} y^{- 5}}{a^{3} y^{- 7}})}^{- 3}$

Step 6

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

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Step 6.1

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

$\frac{y^{6 \cdot 2}}{{(a^{7})}^{2}} {(\frac{a^{4} y^{- 5}}{a^{3} y^{- 7}})}^{- 3}$

Step 6.2

Multiply $6$ by $2$ .

$\frac{y^{12}}{{(a^{7})}^{2}} {(\frac{a^{4} y^{- 5}}{a^{3} y^{- 7}})}^{- 3}$

Step 7

Multiply the exponents in ${(a^{7})}^{2}$ .

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Step 7.1

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

$\frac{y^{12}}{a^{7 \cdot 2}} {(\frac{a^{4} y^{- 5}}{a^{3} y^{- 7}})}^{- 3}$

Step 7.2

Multiply $7$ by $2$ .

$\frac{y^{12}}{a^{14}} {(\frac{a^{4} y^{- 5}}{a^{3} y^{- 7}})}^{- 3}$

Step 8

Move $y^{- 7}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{y^{12}}{a^{14}} {(\frac{a^{4} y^{- 5} y^{7}}{a^{3}})}^{- 3}$

Step 9

Multiply $y^{- 5}$ by $y^{7}$ by adding the exponents.

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Step 9.1

Move $y^{7}$ .

$\frac{y^{12}}{a^{14}} {(\frac{a^{4} (y^{7} y^{- 5})}{a^{3}})}^{- 3}$

Step 9.2

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

$\frac{y^{12}}{a^{14}} {(\frac{a^{4} y^{7 - 5}}{a^{3}})}^{- 3}$

Step 9.3

Subtract $5$ from $7$ .

$\frac{y^{12}}{a^{14}} {(\frac{a^{4} y^{2}}{a^{3}})}^{- 3}$

Step 10

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

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Step 10.1

Factor $a^{3}$ out of $a^{4} y^{2}$ .

$\frac{y^{12}}{a^{14}} {(\frac{a^{3} (a y^{2})}{a^{3}})}^{- 3}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Multiply by $1$ .

$\frac{y^{12}}{a^{14}} {(\frac{a^{3} (a y^{2})}{a^{3} \cdot 1})}^{- 3}$

Step 10.2.2

Cancel the common factor.

$\frac{y^{12}}{a^{14}} {(\frac{a^{3} (a y^{2})}{a^{3} \cdot 1})}^{- 3}$

Step 10.2.3

Rewrite the expression.

$\frac{y^{12}}{a^{14}} {(\frac{a y^{2}}{1})}^{- 3}$

Step 10.2.4

Divide $a y^{2}$ by $1$ .

$\frac{y^{12}}{a^{14}} {(a y^{2})}^{- 3}$

Step 11

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

$\frac{y^{12}}{a^{14}} \cdot \frac{1}{{(a y^{2})}^{3}}$

Step 12

Combine fractions.

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Step 12.1

Combine.

$\frac{y^{12} \cdot 1}{a^{14} {(a y^{2})}^{3}}$

Step 12.2

Multiply $y^{12}$ by $1$ .

$\frac{y^{12}}{a^{14} {(a y^{2})}^{3}}$

Step 13

Simplify the denominator.

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Step 13.1

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

$\frac{y^{12}}{a^{14} a^{3} {(y^{2})}^{3}}$

Step 13.2

Multiply $a^{14}$ by $a^{3}$ by adding the exponents.

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Step 13.2.1

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

$\frac{y^{12}}{a^{14 + 3} {(y^{2})}^{3}}$

Step 13.2.2

Add $14$ and $3$ .

$\frac{y^{12}}{a^{17} {(y^{2})}^{3}}$

Step 13.3

Multiply the exponents in ${(y^{2})}^{3}$ .

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Step 13.3.1

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

$\frac{y^{12}}{a^{17} y^{2 \cdot 3}}$

Step 13.3.2

Multiply $2$ by $3$ .

$\frac{y^{12}}{a^{17} y^{6}}$

Step 14

Cancel the common factor of $y^{12}$ and $y^{6}$ .

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Step 14.1

Factor $y^{6}$ out of $y^{12}$ .

$\frac{y^{6} y^{6}}{a^{17} y^{6}}$

Step 14.2

Cancel the common factors.

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Step 14.2.1

Factor $y^{6}$ out of $a^{17} y^{6}$ .

$\frac{y^{6} y^{6}}{y^{6} a^{17}}$

Step 14.2.2

Cancel the common factor.

$\frac{y^{6} y^{6}}{y^{6} a^{17}}$

Step 14.2.3

Rewrite the expression.

$\frac{y^{6}}{a^{17}}$