Algebra Examples

$\frac{{(x + 5)}^{2}}{x + 6} \cdot \frac{{(x + 6)}^{3}}{x + 5}$

Step 1

Reduce the expression by cancelling the common factors.

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Step 1.1

Cancel the common factor of $x + 5$ .

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Step 1.1.1

Factor $x + 5$ out of ${(x + 5)}^{2}$ .

$\frac{(x + 5) (x + 5)}{x + 6} \cdot \frac{{(x + 6)}^{3}}{x + 5}$

Step 1.1.2

Cancel the common factor.

$\frac{(x + 5) (x + 5)}{x + 6} \cdot \frac{{(x + 6)}^{3}}{x + 5}$

Step 1.1.3

Rewrite the expression.

$\frac{x + 5}{x + 6} \cdot {(x + 6)}^{3}$

Step 1.2

Cancel the common factor of $x + 6$ .

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Step 1.2.1

Factor $x + 6$ out of ${(x + 6)}^{3}$ .

$\frac{x + 5}{x + 6} \cdot ((x + 6) {(x + 6)}^{2})$

Step 1.2.2

Cancel the common factor.

$\frac{x + 5}{x + 6} \cdot ((x + 6) {(x + 6)}^{2})$

Step 1.2.3

Rewrite the expression.

$(x + 5) \cdot {(x + 6)}^{2}$

Step 1.3

Rewrite ${(x + 6)}^{2}$ as $(x + 6) (x + 6)$ .

$(x + 5) \cdot ((x + 6) (x + 6))$

Step 2

Expand $(x + 6) (x + 6)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$(x + 5) \cdot (x (x + 6) + 6 (x + 6))$

Step 2.2

Apply the distributive property.

$(x + 5) \cdot (x \cdot x + x \cdot 6 + 6 (x + 6))$

Step 2.3

Apply the distributive property.

$(x + 5) \cdot (x \cdot x + x \cdot 6 + 6 x + 6 \cdot 6)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

$(x + 5) \cdot (x^{2} + x \cdot 6 + 6 x + 6 \cdot 6)$

Step 3.1.2

Move $6$ to the left of $x$ .

$(x + 5) \cdot (x^{2} + 6 \cdot x + 6 x + 6 \cdot 6)$

Step 3.1.3

Multiply $6$ by $6$ .

$(x + 5) \cdot (x^{2} + 6 x + 6 x + 36)$

Step 3.2

Add $6 x$ and $6 x$ .

$(x + 5) \cdot (x^{2} + 12 x + 36)$

Step 4

Expand $(x + 5) (x^{2} + 12 x + 36)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x^{2} + x (12 x) + x \cdot 36 + 5 x^{2} + 5 (12 x) + 5 \cdot 36$

Step 5

Simplify terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

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

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Step 5.1.1.1

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

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Step 5.1.1.1.1

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

$x^{1} x^{2} + x (12 x) + x \cdot 36 + 5 x^{2} + 5 (12 x) + 5 \cdot 36$

Step 5.1.1.1.2

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

$x^{1 + 2} + x (12 x) + x \cdot 36 + 5 x^{2} + 5 (12 x) + 5 \cdot 36$

Step 5.1.1.2

Add $1$ and $2$ .

$x^{3} + x (12 x) + x \cdot 36 + 5 x^{2} + 5 (12 x) + 5 \cdot 36$

Step 5.1.2

Rewrite using the commutative property of multiplication.

$x^{3} + 12 x \cdot x + x \cdot 36 + 5 x^{2} + 5 (12 x) + 5 \cdot 36$

Step 5.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 5.1.3.1

Move $x$ .

$x^{3} + 12 (x \cdot x) + x \cdot 36 + 5 x^{2} + 5 (12 x) + 5 \cdot 36$

Step 5.1.3.2

Multiply $x$ by $x$ .

$x^{3} + 12 x^{2} + x \cdot 36 + 5 x^{2} + 5 (12 x) + 5 \cdot 36$

Step 5.1.4

Move $36$ to the left of $x$ .

$x^{3} + 12 x^{2} + 36 \cdot x + 5 x^{2} + 5 (12 x) + 5 \cdot 36$

Step 5.1.5

Multiply $12$ by $5$ .

$x^{3} + 12 x^{2} + 36 x + 5 x^{2} + 60 x + 5 \cdot 36$

Step 5.1.6

Multiply $5$ by $36$ .

$x^{3} + 12 x^{2} + 36 x + 5 x^{2} + 60 x + 180$

Step 5.2

Simplify by adding terms.

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Step 5.2.1

Add $12 x^{2}$ and $5 x^{2}$ .

$x^{3} + 17 x^{2} + 36 x + 60 x + 180$

Step 5.2.2

Add $36 x$ and $60 x$ .

$x^{3} + 17 x^{2} + 96 x + 180$