Basic Math Examples

$\frac{{(4 a^{- 4} y^{3})}^{2}}{{(6 a^{- 10} y^{5})}^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(4 a^{- 4})}^{2} {(y^{3})}^{2}}{{(6 a^{- 10} y^{5})}^{3}}$

Step 1.2

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

$\frac{{(4 \frac{1}{a^{4}})}^{2} {(y^{3})}^{2}}{{(6 a^{- 10} y^{5})}^{3}}$

Step 1.3

Combine $4$ and $\frac{1}{a^{4}}$ .

$\frac{{(\frac{4}{a^{4}})}^{2} {(y^{3})}^{2}}{{(6 a^{- 10} y^{5})}^{3}}$

Step 1.4

Apply the product rule to $\frac{4}{a^{4}}$ .

$\frac{\frac{4^{2}}{{(a^{4})}^{2}} {(y^{3})}^{2}}{{(6 a^{- 10} y^{5})}^{3}}$

Step 1.5

Raise $4$ to the power of $2$ .

$\frac{\frac{16}{{(a^{4})}^{2}} {(y^{3})}^{2}}{{(6 a^{- 10} y^{5})}^{3}}$

Step 1.6

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

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Step 1.6.1

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

$\frac{\frac{16}{a^{4 \cdot 2}} {(y^{3})}^{2}}{{(6 a^{- 10} y^{5})}^{3}}$

Step 1.6.2

Multiply $4$ by $2$ .

$\frac{\frac{16}{a^{8}} {(y^{3})}^{2}}{{(6 a^{- 10} y^{5})}^{3}}$

Step 1.7

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

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Step 1.7.1

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

$\frac{\frac{16}{a^{8}} y^{3 \cdot 2}}{{(6 a^{- 10} y^{5})}^{3}}$

Step 1.7.2

Multiply $3$ by $2$ .

$\frac{\frac{16}{a^{8}} y^{6}}{{(6 a^{- 10} y^{5})}^{3}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $6 a^{- 10} y^{5}$ .

$\frac{\frac{16}{a^{8}} y^{6}}{{(6 a^{- 10})}^{3} {(y^{5})}^{3}}$

Step 2.2

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

$\frac{\frac{16}{a^{8}} y^{6}}{{(6 \frac{1}{a^{10}})}^{3} {(y^{5})}^{3}}$

Step 2.3

Combine $6$ and $\frac{1}{a^{10}}$ .

$\frac{\frac{16}{a^{8}} y^{6}}{{(\frac{6}{a^{10}})}^{3} {(y^{5})}^{3}}$

Step 2.4

Apply the product rule to $\frac{6}{a^{10}}$ .

$\frac{\frac{16}{a^{8}} y^{6}}{\frac{6^{3}}{{(a^{10})}^{3}} {(y^{5})}^{3}}$

Step 2.5

Raise $6$ to the power of $3$ .

$\frac{\frac{16}{a^{8}} y^{6}}{\frac{216}{{(a^{10})}^{3}} {(y^{5})}^{3}}$

Step 2.6

Multiply the exponents in ${(a^{10})}^{3}$ .

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Step 2.6.1

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

$\frac{\frac{16}{a^{8}} y^{6}}{\frac{216}{a^{10 \cdot 3}} {(y^{5})}^{3}}$

Step 2.6.2

Multiply $10$ by $3$ .

$\frac{\frac{16}{a^{8}} y^{6}}{\frac{216}{a^{30}} {(y^{5})}^{3}}$

Step 2.7

Multiply the exponents in ${(y^{5})}^{3}$ .

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Step 2.7.1

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

$\frac{\frac{16}{a^{8}} y^{6}}{\frac{216}{a^{30}} y^{5 \cdot 3}}$

Step 2.7.2

Multiply $5$ by $3$ .

$\frac{\frac{16}{a^{8}} y^{6}}{\frac{216}{a^{30}} y^{15}}$

Step 3

Combine fractions.

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Step 3.1

Combine $\frac{16}{a^{8}}$ and $y^{6}$ .

$\frac{\frac{16 y^{6}}{a^{8}}}{\frac{216}{a^{30}} y^{15}}$

Step 3.2

Combine $\frac{216}{a^{30}}$ and $y^{15}$ .

$\frac{\frac{16 y^{6}}{a^{8}}}{\frac{216 y^{15}}{a^{30}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{16 y^{6}}{a^{8}} \cdot \frac{a^{30}}{216 y^{15}}$

Step 5

Combine.

$\frac{16 y^{6} a^{30}}{a^{8} (216 y^{15})}$

Step 6

Cancel the common factor of $16$ and $216$ .

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Step 6.1

Factor $8$ out of $16 y^{6} a^{30}$ .

$\frac{8 (2 y^{6} a^{30})}{a^{8} (216 y^{15})}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $8$ out of $a^{8} (216 y^{15})$ .

$\frac{8 (2 y^{6} a^{30})}{8 (a^{8} (27 y^{15}))}$

Step 6.2.2

Cancel the common factor.

$\frac{8 (2 y^{6} a^{30})}{8 (a^{8} (27 y^{15}))}$

Step 6.2.3

Rewrite the expression.

$\frac{2 y^{6} a^{30}}{a^{8} (27 y^{15})}$

Step 7

Cancel the common factor of $y^{6}$ and $y^{15}$ .

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Step 7.1

Factor $y^{6}$ out of $2 y^{6} a^{30}$ .

$\frac{y^{6} (2 a^{30})}{a^{8} (27 y^{15})}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $y^{6}$ out of $a^{8} (27 y^{15})$ .

$\frac{y^{6} (2 a^{30})}{y^{6} (a^{8} (27 y^{9}))}$

Step 7.2.2

Cancel the common factor.

$\frac{y^{6} (2 a^{30})}{y^{6} (a^{8} (27 y^{9}))}$

Step 7.2.3

Rewrite the expression.

$\frac{2 a^{30}}{a^{8} (27 y^{9})}$

Step 8

Cancel the common factor of $a^{30}$ and $a^{8}$ .

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Step 8.1

Factor $a^{8}$ out of $2 a^{30}$ .

$\frac{a^{8} (2 a^{22})}{a^{8} (27 y^{9})}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Cancel the common factor.

$\frac{a^{8} (2 a^{22})}{a^{8} (27 y^{9})}$

Step 8.2.2

Rewrite the expression.

$\frac{2 a^{22}}{27 y^{9}}$