Basic Math Examples

${(\frac{- 16 a^{5} b^{7}}{(2 a^{- 2} b) (a b^{- 2})})}^{2}$

Step 1

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

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Step 1.1

Move $a$ .

${(\frac{- 16 a^{5} b^{7}}{2 (a \cdot a^{- 2}) b \cdot b^{- 2}})}^{2}$

Step 1.2

Multiply $a$ by $a^{- 2}$ .

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Step 1.2.1

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

${(\frac{- 16 a^{5} b^{7}}{2 (a^{1} a^{- 2}) b \cdot b^{- 2}})}^{2}$

Step 1.2.2

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

${(\frac{- 16 a^{5} b^{7}}{2 a^{1 - 2} b \cdot b^{- 2}})}^{2}$

Step 1.3

Subtract $2$ from $1$ .

${(\frac{- 16 a^{5} b^{7}}{2 a^{- 1} b \cdot b^{- 2}})}^{2}$

Step 2

Multiply $b$ by $b^{- 2}$ by adding the exponents.

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Step 2.1

Move $b^{- 2}$ .

${(\frac{- 16 a^{5} b^{7}}{2 a^{- 1} (b^{- 2} b)})}^{2}$

Step 2.2

Multiply $b^{- 2}$ by $b$ .

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Step 2.2.1

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

${(\frac{- 16 a^{5} b^{7}}{2 a^{- 1} (b^{- 2} b^{1})})}^{2}$

Step 2.2.2

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

${(\frac{- 16 a^{5} b^{7}}{2 a^{- 1} b^{- 2 + 1}})}^{2}$

Step 2.3

Add $- 2$ and $1$ .

${(\frac{- 16 a^{5} b^{7}}{2 a^{- 1} b^{- 1}})}^{2}$

Step 3

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

${(\frac{- 16 a^{5} b^{7} a}{2 b^{- 1}})}^{2}$

Step 4

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

${(\frac{- 16 a^{5} b^{7} a b}{2})}^{2}$

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Move $a$ .

${(\frac{- 16 (a \cdot a^{5}) b^{7} b}{2})}^{2}$

Step 5.1.2

Multiply $a$ by $a^{5}$ .

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Step 5.1.2.1

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

${(\frac{- 16 (a^{1} a^{5}) b^{7} b}{2})}^{2}$

Step 5.1.2.2

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

${(\frac{- 16 a^{1 + 5} b^{7} b}{2})}^{2}$

Step 5.1.3

Add $1$ and $5$ .

${(\frac{- 16 a^{6} b^{7} b}{2})}^{2}$

Step 5.2

Multiply $b^{7}$ by $b$ by adding the exponents.

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Step 5.2.1

Move $b$ .

${(\frac{- 16 a^{6} (b \cdot b^{7})}{2})}^{2}$

Step 5.2.2

Multiply $b$ by $b^{7}$ .

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Step 5.2.2.1

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

${(\frac{- 16 a^{6} (b^{1} b^{7})}{2})}^{2}$

Step 5.2.2.2

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

${(\frac{- 16 a^{6} b^{1 + 7}}{2})}^{2}$

Step 5.2.3

Add $1$ and $7$ .

${(\frac{- 16 a^{6} b^{8}}{2})}^{2}$

Step 6

Cancel the common factor of $- 16$ and $2$ .

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Step 6.1

Factor $2$ out of $- 16 a^{6} b^{8}$ .

${(\frac{2 (- 8 a^{6} b^{8})}{2})}^{2}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $2$ out of $2$ .

${(\frac{2 (- 8 a^{6} b^{8})}{2 (1)})}^{2}$

Step 6.2.2

Cancel the common factor.

${(\frac{2 (- 8 a^{6} b^{8})}{2 \cdot 1})}^{2}$

Step 6.2.3

Rewrite the expression.

${(\frac{- 8 a^{6} b^{8}}{1})}^{2}$

Step 6.2.4

Divide $- 8 a^{6} b^{8}$ by $1$ .

${(- 8 a^{6} b^{8})}^{2}$

Step 7

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

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Step 7.1

Apply the product rule to $- 8 a^{6} b^{8}$ .

${(- 8 a^{6})}^{2} {(b^{8})}^{2}$

Step 7.2

Apply the product rule to $- 8 a^{6}$ .

${(- 8)}^{2} {(a^{6})}^{2} {(b^{8})}^{2}$

Step 8

Raise $- 8$ to the power of $2$ .

$64 {(a^{6})}^{2} {(b^{8})}^{2}$

Step 9

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

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Step 9.1

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

$64 a^{6 \cdot 2} {(b^{8})}^{2}$

Step 9.2

Multiply $6$ by $2$ .

$64 a^{12} {(b^{8})}^{2}$

Step 10

Multiply the exponents in ${(b^{8})}^{2}$ .

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Step 10.1

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

$64 a^{12} b^{8 \cdot 2}$

Step 10.2

Multiply $8$ by $2$ .

$64 a^{12} b^{16}$