Basic Math Examples

${(\frac{2^{- 3} a y^{- 4}}{a^{- 4} y})}^{4} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 1

Move $2^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{a y^{- 4}}{a^{- 4} y \cdot 2^{3}})}^{4} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 2

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

${(\frac{a}{a^{- 4} y \cdot 2^{3} y^{4}})}^{4} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 3

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

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Step 3.1

Move $y^{4}$ .

${(\frac{a}{a^{- 4} (y^{4} y) \cdot 2^{3}})}^{4} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 3.2

Multiply $y^{4}$ by $y$ .

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Step 3.2.1

Raise $y$ to the power of $1$ .

${(\frac{a}{a^{- 4} (y^{4} y^{1}) \cdot 2^{3}})}^{4} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 3.2.2

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

${(\frac{a}{a^{- 4} y^{4 + 1} \cdot 2^{3}})}^{4} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 3.3

Add $4$ and $1$ .

${(\frac{a}{a^{- 4} y^{5} \cdot 2^{3}})}^{4} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 4

Move $a^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{a \cdot a^{4}}{y^{5} \cdot 2^{3}})}^{4} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 5

Multiply $a$ by $a^{4}$ by adding the exponents.

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Step 5.1

Multiply $a$ by $a^{4}$ .

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Step 5.1.1

Raise $a$ to the power of $1$ .

${(\frac{a^{1} a^{4}}{y^{5} \cdot 2^{3}})}^{4} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 5.1.2

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

${(\frac{a^{1 + 4}}{y^{5} \cdot 2^{3}})}^{4} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 5.2

Add $1$ and $4$ .

${(\frac{a^{5}}{y^{5} \cdot 2^{3}})}^{4} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 6

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

${(\frac{a^{5}}{y^{5} \cdot 8})}^{4} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 7

Move $8$ to the left of $y^{5}$ .

${(\frac{a^{5}}{8 \cdot y^{5}})}^{4} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 8

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.1

Apply the product rule to $\frac{a^{5}}{8 y^{5}}$ .

$\frac{{(a^{5})}^{4}}{{(8 y^{5})}^{4}} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 8.2

Apply the product rule to $8 y^{5}$ .

$\frac{{(a^{5})}^{4}}{8^{4} {(y^{5})}^{4}} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 9

Multiply the exponents in ${(a^{5})}^{4}$ .

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Step 9.1

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

$\frac{a^{5 \cdot 4}}{8^{4} {(y^{5})}^{4}} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 9.2

Multiply $5$ by $4$ .

$\frac{a^{20}}{8^{4} {(y^{5})}^{4}} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 10

Simplify the denominator.

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Step 10.1

Raise $8$ to the power of $4$ .

$\frac{a^{20}}{4096 {(y^{5})}^{4}} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 10.2

Multiply the exponents in ${(y^{5})}^{4}$ .

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Step 10.2.1

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

$\frac{a^{20}}{4096 y^{5 \cdot 4}} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 10.2.2

Multiply $5$ by $4$ .

$\frac{a^{20}}{4096 y^{20}} {(\frac{8^{- 1} a^{- 3} y}{a^{8} y^{- 1}})}^{- 1}$

Step 11

Simplify the expression.

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Step 11.1

Move $8^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{a^{20}}{4096 y^{20}} {(\frac{a^{- 3} y}{a^{8} y^{- 1} \cdot 8})}^{- 1}$

Step 11.2

Move $a^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{a^{20}}{4096 y^{20}} {(\frac{y}{a^{8} y^{- 1} \cdot 8 a^{3}})}^{- 1}$

Step 12

Multiply $a^{8}$ by $a^{3}$ by adding the exponents.

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Step 12.1

Move $a^{3}$ .

$\frac{a^{20}}{4096 y^{20}} {(\frac{y}{a^{3} a^{8} y^{- 1} \cdot 8})}^{- 1}$

Step 12.2

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

$\frac{a^{20}}{4096 y^{20}} {(\frac{y}{a^{3 + 8} y^{- 1} \cdot 8})}^{- 1}$

Step 12.3

Add $3$ and $8$ .

$\frac{a^{20}}{4096 y^{20}} {(\frac{y}{a^{11} y^{- 1} \cdot 8})}^{- 1}$

Step 13

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

$\frac{a^{20}}{4096 y^{20}} {(\frac{y \cdot y}{a^{11} \cdot 8})}^{- 1}$

Step 14

Multiply $y$ by $y$ .

$\frac{a^{20}}{4096 y^{20}} {(\frac{y^{2}}{a^{11} \cdot 8})}^{- 1}$

Step 15

Move $8$ to the left of $a^{11}$ .

$\frac{a^{20}}{4096 y^{20}} {(\frac{y^{2}}{8 \cdot a^{11}})}^{- 1}$

Step 16

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{a^{20}}{4096 y^{20}} \cdot \frac{8 a^{11}}{y^{2}}$

Step 17

Combine.

$\frac{a^{20} (8 a^{11})}{4096 y^{20} y^{2}}$

Step 18

Multiply $a^{20}$ by $a^{11}$ by adding the exponents.

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Step 18.1

Move $a^{11}$ .

$\frac{a^{11} a^{20} \cdot 8}{4096 y^{20} y^{2}}$

Step 18.2

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

$\frac{a^{11 + 20} \cdot 8}{4096 y^{20} y^{2}}$

Step 18.3

Add $11$ and $20$ .

$\frac{a^{31} \cdot 8}{4096 y^{20} y^{2}}$

Step 19

Multiply $y^{20}$ by $y^{2}$ by adding the exponents.

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Step 19.1

Move $y^{2}$ .

$\frac{a^{31} \cdot 8}{4096 (y^{2} y^{20})}$

Step 19.2

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

$\frac{a^{31} \cdot 8}{4096 y^{2 + 20}}$

Step 19.3

Add $2$ and $20$ .

$\frac{a^{31} \cdot 8}{4096 y^{22}}$

Step 20

Cancel the common factor of $8$ and $4096$ .

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Step 20.1

Factor $8$ out of $a^{31} \cdot 8$ .

$\frac{8 \cdot a^{31}}{4096 y^{22}}$

Step 20.2

Cancel the common factors.

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Step 20.2.1

Factor $8$ out of $4096 y^{22}$ .

$\frac{8 \cdot a^{31}}{8 (512 y^{22})}$

Step 20.2.2

Cancel the common factor.

$\frac{8 \cdot a^{31}}{8 (512 y^{22})}$

Step 20.2.3

Rewrite the expression.

$\frac{a^{31}}{512 y^{22}}$