Algebra Examples

${(\frac{1}{4} p + \frac{3}{4} q)}^{3}$

Step 1

Simplify each term.

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Step 1.1

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

${(\frac{p}{4} + \frac{3}{4} q)}^{3}$

Step 1.2

Combine $\frac{3}{4}$ and $q$ .

${(\frac{p}{4} + \frac{3 q}{4})}^{3}$

Step 2

Use the Binomial Theorem.

${(\frac{p}{4})}^{3} + 3 {(\frac{p}{4})}^{2} \frac{3 q}{4} + 3 \frac{p}{4} {(\frac{3 q}{4})}^{2} + {(\frac{3 q}{4})}^{3}$

Step 3

Simplify each term.

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Step 3.1

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

$\frac{p^{3}}{4^{3}} + 3 {(\frac{p}{4})}^{2} \frac{3 q}{4} + 3 \frac{p}{4} {(\frac{3 q}{4})}^{2} + {(\frac{3 q}{4})}^{3}$

Step 3.2

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

$\frac{p^{3}}{64} + 3 {(\frac{p}{4})}^{2} \frac{3 q}{4} + 3 \frac{p}{4} {(\frac{3 q}{4})}^{2} + {(\frac{3 q}{4})}^{3}$

Step 3.3

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

$\frac{p^{3}}{64} + 3 \frac{p^{2}}{4^{2}} \frac{3 q}{4} + 3 \frac{p}{4} {(\frac{3 q}{4})}^{2} + {(\frac{3 q}{4})}^{3}$

Step 3.4

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

$\frac{p^{3}}{64} + 3 \frac{p^{2}}{16} \frac{3 q}{4} + 3 \frac{p}{4} {(\frac{3 q}{4})}^{2} + {(\frac{3 q}{4})}^{3}$

Step 3.5

Combine $3$ and $\frac{p^{2}}{16}$ .

$\frac{p^{3}}{64} + \frac{3 p^{2}}{16} \cdot \frac{3 q}{4} + 3 \frac{p}{4} {(\frac{3 q}{4})}^{2} + {(\frac{3 q}{4})}^{3}$

Step 3.6

Combine.

$\frac{p^{3}}{64} + \frac{3 p^{2} (3 q)}{16 \cdot 4} + 3 \frac{p}{4} {(\frac{3 q}{4})}^{2} + {(\frac{3 q}{4})}^{3}$

Step 3.7

Multiply $3$ by $3$ .

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{16 \cdot 4} + 3 \frac{p}{4} {(\frac{3 q}{4})}^{2} + {(\frac{3 q}{4})}^{3}$

Step 3.8

Multiply $16$ by $4$ .

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + 3 \frac{p}{4} {(\frac{3 q}{4})}^{2} + {(\frac{3 q}{4})}^{3}$

Step 3.9

Combine $3$ and $\frac{p}{4}$ .

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{3 p}{4} {(\frac{3 q}{4})}^{2} + {(\frac{3 q}{4})}^{3}$

Step 3.10

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

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Step 3.10.1

Apply the product rule to $\frac{3 q}{4}$ .

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{3 p}{4} \cdot \frac{{(3 q)}^{2}}{4^{2}} + {(\frac{3 q}{4})}^{3}$

Step 3.10.2

Apply the product rule to $3 q$ .

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{3 p}{4} \cdot \frac{3^{2} q^{2}}{4^{2}} + {(\frac{3 q}{4})}^{3}$

Step 3.11

Combine.

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{3 p (3^{2} q^{2})}{4 \cdot 4^{2}} + {(\frac{3 q}{4})}^{3}$

Step 3.12

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

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Step 3.12.1

Move $3^{2}$ .

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{3^{2} \cdot 3 p q^{2}}{4 \cdot 4^{2}} + {(\frac{3 q}{4})}^{3}$

Step 3.12.2

Multiply $3^{2}$ by $3$ .

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Step 3.12.2.1

Raise $3$ to the power of $1$ .

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{3^{2} \cdot 3^{1} p q^{2}}{4 \cdot 4^{2}} + {(\frac{3 q}{4})}^{3}$

Step 3.12.2.2

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

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{3^{2 + 1} p q^{2}}{4 \cdot 4^{2}} + {(\frac{3 q}{4})}^{3}$

Step 3.12.3

Add $2$ and $1$ .

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{3^{3} p q^{2}}{4 \cdot 4^{2}} + {(\frac{3 q}{4})}^{3}$

Step 3.13

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

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Step 3.13.1

Multiply $4$ by $4^{2}$ .

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Step 3.13.1.1

Raise $4$ to the power of $1$ .

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{3^{3} p q^{2}}{4^{1} \cdot 4^{2}} + {(\frac{3 q}{4})}^{3}$

Step 3.13.1.2

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

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{3^{3} p q^{2}}{4^{1 + 2}} + {(\frac{3 q}{4})}^{3}$

Step 3.13.2

Add $1$ and $2$ .

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{3^{3} p q^{2}}{4^{3}} + {(\frac{3 q}{4})}^{3}$

Step 3.14

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

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{27 p q^{2}}{4^{3}} + {(\frac{3 q}{4})}^{3}$

Step 3.15

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

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{27 p q^{2}}{64} + {(\frac{3 q}{4})}^{3}$

Step 3.16

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

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Step 3.16.1

Apply the product rule to $\frac{3 q}{4}$ .

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{27 p q^{2}}{64} + \frac{{(3 q)}^{3}}{4^{3}}$

Step 3.16.2

Apply the product rule to $3 q$ .

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{27 p q^{2}}{64} + \frac{3^{3} q^{3}}{4^{3}}$

Step 3.17

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

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{27 p q^{2}}{64} + \frac{27 q^{3}}{4^{3}}$

Step 3.18

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

$\frac{p^{3}}{64} + \frac{9 p^{2} q}{64} + \frac{27 p q^{2}}{64} + \frac{27 q^{3}}{64}$