Pre-Algebra Examples

$\frac{{(4 p)}^{2} p^{3}}{p^{- 1} p^{- 5}} \cdot {(4 p^{- 4})}^{- 3}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $4 p$ .

$\frac{4^{2} p^{2} p^{3}}{p^{- 1} p^{- 5}} \cdot {(4 p^{- 4})}^{- 3}$

Step 1.2

Multiply $p^{2}$ by $p^{3}$ by adding the exponents.

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Step 1.2.1

Move $p^{3}$ .

$\frac{4^{2} (p^{3} p^{2})}{p^{- 1} p^{- 5}} \cdot {(4 p^{- 4})}^{- 3}$

Step 1.2.2

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

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

Step 1.2.3

Add $3$ and $2$ .

$\frac{4^{2} p^{5}}{p^{- 1} p^{- 5}} \cdot {(4 p^{- 4})}^{- 3}$

Step 1.3

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

$\frac{16 p^{5}}{p^{- 1} p^{- 5}} \cdot {(4 p^{- 4})}^{- 3}$

Step 2

Multiply $p^{- 1}$ by $p^{- 5}$ by adding the exponents.

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Step 2.1

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

$\frac{16 p^{5}}{p^{- 1 - 5}} \cdot {(4 p^{- 4})}^{- 3}$

Step 2.2

Subtract $5$ from $- 1$ .

$\frac{16 p^{5}}{p^{- 6}} \cdot {(4 p^{- 4})}^{- 3}$

Step 3

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

$\frac{16 p^{5}}{\frac{1}{p^{6}}} \cdot {(4 p^{- 4})}^{- 3}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$16 p^{5} p^{6} \cdot {(4 p^{- 4})}^{- 3}$

Step 5

Multiply $p^{5}$ by $p^{6}$ by adding the exponents.

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Step 5.1

Move $p^{6}$ .

$16 (p^{6} p^{5}) \cdot {(4 p^{- 4})}^{- 3}$

Step 5.2

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

$16 p^{6 + 5} \cdot {(4 p^{- 4})}^{- 3}$

Step 5.3

Add $6$ and $5$ .

$16 p^{11} \cdot {(4 p^{- 4})}^{- 3}$

Step 6

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

$16 p^{11} \cdot {(4 \frac{1}{p^{4}})}^{- 3}$

Step 7

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

$16 p^{11} \cdot {(\frac{4}{p^{4}})}^{- 3}$

Step 8

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

$16 p^{11} \cdot {(\frac{p^{4}}{4})}^{3}$

Step 9

Simplify the expression.

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Step 9.1

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

$16 p^{11} \cdot \frac{{(p^{4})}^{3}}{4^{3}}$

Step 9.2

Multiply the exponents in ${(p^{4})}^{3}$ .

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Step 9.2.1

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

$16 p^{11} \cdot \frac{p^{4 \cdot 3}}{4^{3}}$

Step 9.2.2

Multiply $4$ by $3$ .

$16 p^{11} \cdot \frac{p^{12}}{4^{3}}$

Step 9.3

Raise $4$ to the power of $3$ .

$16 p^{11} \cdot \frac{p^{12}}{64}$

Step 10

Cancel the common factor of $16$ .

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Step 10.1

Factor $16$ out of $16 p^{11}$ .

$16 (p^{11}) \cdot \frac{p^{12}}{64}$

Step 10.2

Factor $16$ out of $64$ .

$16 (p^{11}) \cdot \frac{p^{12}}{16 (4)}$

Step 10.3

Cancel the common factor.

$16 p^{11} \cdot \frac{p^{12}}{16 \cdot 4}$

Step 10.4

Rewrite the expression.

$p^{11} \cdot \frac{p^{12}}{4}$

Step 11

Combine $p^{11}$ and $\frac{p^{12}}{4}$ .

$\frac{p^{11} p^{12}}{4}$

Step 12

Multiply $p^{11}$ by $p^{12}$ by adding the exponents.

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Step 12.1

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

$\frac{p^{11 + 12}}{4}$

Step 12.2

Add $11$ and $12$ .

$\frac{p^{23}}{4}$