Algebra Examples

${(- 1 \frac{1}{3} p + 6 q)}^{2}$

Step 1

Convert $1 \frac{1}{3}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

${(- (1 + \frac{1}{3}) p + 6 q)}^{2}$

Step 1.2

Add $1$ and $\frac{1}{3}$ .

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

${(- (\frac{3}{3} + \frac{1}{3}) p + 6 q)}^{2}$

Step 1.2.2

Combine the numerators over the common denominator.

${(- \frac{3 + 1}{3} p + 6 q)}^{2}$

Step 1.2.3

Add $3$ and $1$ .

${(- \frac{4}{3} p + 6 q)}^{2}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

${(- \frac{p \cdot 4}{3} + 6 q)}^{2}$

Step 2.1.2

Move $4$ to the left of $p$ .

${(- \frac{4 p}{3} + 6 q)}^{2}$

Step 2.2

Rewrite ${(- \frac{4 p}{3} + 6 q)}^{2}$ as $(- \frac{4 p}{3} + 6 q) (- \frac{4 p}{3} + 6 q)$ .

$(- \frac{4 p}{3} + 6 q) (- \frac{4 p}{3} + 6 q)$

Step 3

Expand $(- \frac{4 p}{3} + 6 q) (- \frac{4 p}{3} + 6 q)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$- \frac{4 p}{3} (- \frac{4 p}{3} + 6 q) + 6 q (- \frac{4 p}{3} + 6 q)$

Step 3.2

Apply the distributive property.

$- \frac{4 p}{3} (- \frac{4 p}{3}) - \frac{4 p}{3} (6 q) + 6 q (- \frac{4 p}{3} + 6 q)$

Step 3.3

Apply the distributive property.

$- \frac{4 p}{3} (- \frac{4 p}{3}) - \frac{4 p}{3} (6 q) + 6 q (- \frac{4 p}{3}) + 6 q (6 q)$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $- \frac{4 p}{3} (- \frac{4 p}{3})$ .

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Step 4.1.1.1

Multiply $- 1$ by $- 1$ .

$1 \frac{4 p}{3} \frac{4 p}{3} - \frac{4 p}{3} (6 q) + 6 q (- \frac{4 p}{3}) + 6 q (6 q)$

Step 4.1.1.2

Multiply $\frac{4 p}{3}$ by $1$ .

$\frac{4 p}{3} \cdot \frac{4 p}{3} - \frac{4 p}{3} (6 q) + 6 q (- \frac{4 p}{3}) + 6 q (6 q)$

Step 4.1.1.3

Multiply $\frac{4 p}{3}$ by $\frac{4 p}{3}$ .

$\frac{4 p (4 p)}{3 \cdot 3} - \frac{4 p}{3} (6 q) + 6 q (- \frac{4 p}{3}) + 6 q (6 q)$

Step 4.1.1.4

Multiply $4$ by $4$ .

$\frac{16 p \cdot p}{3 \cdot 3} - \frac{4 p}{3} (6 q) + 6 q (- \frac{4 p}{3}) + 6 q (6 q)$

Step 4.1.1.5

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

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

Step 4.1.1.6

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

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

Step 4.1.1.7

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

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

Step 4.1.1.8

Add $1$ and $1$ .

$\frac{16 p^{2}}{3 \cdot 3} - \frac{4 p}{3} (6 q) + 6 q (- \frac{4 p}{3}) + 6 q (6 q)$

Step 4.1.1.9

Multiply $3$ by $3$ .

$\frac{16 p^{2}}{9} - \frac{4 p}{3} (6 q) + 6 q (- \frac{4 p}{3}) + 6 q (6 q)$

Step 4.1.2

Cancel the common factor of $3$ .

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Step 4.1.2.1

Move the leading negative in $- \frac{4 p}{3}$ into the numerator.

$\frac{16 p^{2}}{9} + \frac{- 4 p}{3} (6 q) + 6 q (- \frac{4 p}{3}) + 6 q (6 q)$

Step 4.1.2.2

Factor $3$ out of $6 q$ .

$\frac{16 p^{2}}{9} + \frac{- 4 p}{3} (3 (2 q)) + 6 q (- \frac{4 p}{3}) + 6 q (6 q)$

Step 4.1.2.3

Cancel the common factor.

$\frac{16 p^{2}}{9} + \frac{- 4 p}{3} (3 (2 q)) + 6 q (- \frac{4 p}{3}) + 6 q (6 q)$

Step 4.1.2.4

Rewrite the expression.

$\frac{16 p^{2}}{9} - 4 p (2 q) + 6 q (- \frac{4 p}{3}) + 6 q (6 q)$

Step 4.1.3

Multiply $2$ by $- 4$ .

$\frac{16 p^{2}}{9} - 8 p q + 6 q (- \frac{4 p}{3}) + 6 q (6 q)$

Step 4.1.4

Cancel the common factor of $3$ .

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Step 4.1.4.1

Move the leading negative in $- \frac{4 p}{3}$ into the numerator.

$\frac{16 p^{2}}{9} - 8 p q + 6 q \frac{- 4 p}{3} + 6 q (6 q)$

Step 4.1.4.2

Factor $3$ out of $6 q$ .

$\frac{16 p^{2}}{9} - 8 p q + 3 (2 q) \frac{- 4 p}{3} + 6 q (6 q)$

Step 4.1.4.3

Cancel the common factor.

$\frac{16 p^{2}}{9} - 8 p q + 3 (2 q) \frac{- 4 p}{3} + 6 q (6 q)$

Step 4.1.4.4

Rewrite the expression.

$\frac{16 p^{2}}{9} - 8 p q + 2 q (- 4 p) + 6 q (6 q)$

Step 4.1.5

Multiply $- 4$ by $2$ .

$\frac{16 p^{2}}{9} - 8 p q - 8 q p + 6 q (6 q)$

Step 4.1.6

Rewrite using the commutative property of multiplication.

$\frac{16 p^{2}}{9} - 8 p q - 8 q p + 6 \cdot 6 q \cdot q$

Step 4.1.7

Multiply $q$ by $q$ by adding the exponents.

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Step 4.1.7.1

Move $q$ .

$\frac{16 p^{2}}{9} - 8 p q - 8 q p + 6 \cdot 6 (q \cdot q)$

Step 4.1.7.2

Multiply $q$ by $q$ .

$\frac{16 p^{2}}{9} - 8 p q - 8 q p + 6 \cdot 6 q^{2}$

Step 4.1.8

Multiply $6$ by $6$ .

$\frac{16 p^{2}}{9} - 8 p q - 8 q p + 36 q^{2}$

Step 4.2

Subtract $8 q p$ from $- 8 p q$ .

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Step 4.2.1

Move $q$ .

$\frac{16 p^{2}}{9} - 8 p q - 8 p q + 36 q^{2}$

Step 4.2.2

Subtract $8 p q$ from $- 8 p q$ .

$\frac{16 p^{2}}{9} - 16 p q + 36 q^{2}$