Trigonometry Examples

$\frac{{(b^{2 n + 1})}^{3}}{b^{n} \cdot b^{4 n + 3}}$

Step 1

Multiply $b^{n}$ by $b^{4 n + 3}$ by adding the exponents.

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Step 1.1

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

$\frac{{(b^{2 n + 1})}^{3}}{b^{n + 4 n + 3}}$

Step 1.2

Add $n$ and $4 n$ .

$\frac{{(b^{2 n + 1})}^{3}}{b^{5 n + 3}}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Multiply the exponents in ${(b^{2 n + 1})}^{3}$ .

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Step 2.1.1

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

$\frac{b^{(2 n + 1) \cdot 3}}{b^{5 n + 3}}$

Step 2.1.2

Apply the distributive property.

$\frac{b^{2 n \cdot 3 + 1 \cdot 3}}{b^{5 n + 3}}$

Step 2.1.3

Multiply $3$ by $2$ .

$\frac{b^{6 n + 1 \cdot 3}}{b^{5 n + 3}}$

Step 2.1.4

Multiply $3$ by $1$ .

$\frac{b^{6 n + 3}}{b^{5 n + 3}}$

Step 2.2

Cancel the common factor of $b^{6 n + 3}$ and $b^{5 n + 3}$ .

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Step 2.2.1

Factor $b^{5 n + 3}$ out of $b^{6 n + 3}$ .

$\frac{b^{5 n + 3} b^{6 n + 3 - (5 n + 3)}}{b^{5 n + 3}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Multiply by $1$ .

$\frac{b^{5 n + 3} b^{6 n + 3 - (5 n + 3)}}{b^{5 n + 3} \cdot 1}$

Step 2.2.2.2

Cancel the common factor.

$\frac{b^{5 n + 3} b^{6 n + 3 - (5 n + 3)}}{b^{5 n + 3} \cdot 1}$

Step 2.2.2.3

Rewrite the expression.

$\frac{b^{6 n + 3 - (5 n + 3)}}{1}$

Step 2.2.2.4

Divide $b^{6 n + 3 - (5 n + 3)}$ by $1$ .

$b^{6 n + 3 - (5 n + 3)}$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$b^{6 n + 3 - (5 n) - 1 \cdot 3}$

Step 3.2

Multiply $5$ by $- 1$ .

$b^{6 n + 3 - 5 n - 1 \cdot 3}$

Step 3.3

Multiply $- 1$ by $3$ .

$b^{6 n + 3 - 5 n - 3}$

Step 4

Simplify by adding terms.

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Step 4.1

Combine the opposite terms in $6 n + 3 - 5 n - 3$ .

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Step 4.1.1

Subtract $3$ from $3$ .

$b^{6 n - 5 n + 0}$

Step 4.1.2

Add $6 n - 5 n$ and $0$ .

$b^{6 n - 5 n}$

Step 4.2

Subtract $5 n$ from $6 n$ .

$b^{n}$