Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $5^{2 n - 1} \cdot 2^{n + 1}$ .

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

Step 1.2

Multiply the exponents in ${(5^{2 n - 1})}^{3}$ .

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Step 1.2.1

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

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Multiply $3$ by $2$ .

$\frac{5 \cdot 2^{n} \cdot 5^{6 n - 1 \cdot 3} \cdot {(2^{n + 1})}^{3}}{{(5^{3 n - 1} \cdot 2^{2 n - 1})}^{2}}$

Step 1.2.4

Multiply $- 1$ by $3$ .

$\frac{5 \cdot 2^{n} \cdot 5^{6 n - 3} \cdot {(2^{n + 1})}^{3}}{{(5^{3 n - 1} \cdot 2^{2 n - 1})}^{2}}$

Step 1.3

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

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Step 1.3.1

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

$\frac{5 \cdot 2^{n} \cdot 5^{6 n - 3} \cdot 2^{(n + 1) \cdot 3}}{{(5^{3 n - 1} \cdot 2^{2 n - 1})}^{2}}$

Step 1.3.2

Apply the distributive property.

$\frac{5 \cdot 2^{n} \cdot 5^{6 n - 3} \cdot 2^{n \cdot 3 + 1 \cdot 3}}{{(5^{3 n - 1} \cdot 2^{2 n - 1})}^{2}}$

Step 1.3.3

Move $3$ to the left of $n$ .

$\frac{5 \cdot 2^{n} \cdot 5^{6 n - 3} \cdot 2^{3 \cdot n + 1 \cdot 3}}{{(5^{3 n - 1} \cdot 2^{2 n - 1})}^{2}}$

Step 1.3.4

Multiply $3$ by $1$ .

$\frac{5 \cdot 2^{n} \cdot 5^{6 n - 3} \cdot 2^{3 n + 3}}{{(5^{3 n - 1} \cdot 2^{2 n - 1})}^{2}}$

Step 1.4

Combine exponents.

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Step 1.4.1

Multiply $5$ by $5^{6 n - 3}$ by adding the exponents.

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Step 1.4.1.1

Move $5^{6 n - 3}$ .

$\frac{5^{6 n - 3} \cdot 5 \cdot 2^{n} \cdot 2^{3 n + 3}}{{(5^{3 n - 1} \cdot 2^{2 n - 1})}^{2}}$

Step 1.4.1.2

Multiply $5^{6 n - 3}$ by $5$ .

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Step 1.4.1.2.1

Raise $5$ to the power of $1$ .

$\frac{5^{6 n - 3} \cdot 5^{1} \cdot 2^{n} \cdot 2^{3 n + 3}}{{(5^{3 n - 1} \cdot 2^{2 n - 1})}^{2}}$

Step 1.4.1.2.2

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

$\frac{5^{6 n - 3 + 1} \cdot 2^{n} \cdot 2^{3 n + 3}}{{(5^{3 n - 1} \cdot 2^{2 n - 1})}^{2}}$

Step 1.4.1.3

Add $- 3$ and $1$ .

$\frac{5^{6 n - 2} \cdot 2^{n} \cdot 2^{3 n + 3}}{{(5^{3 n - 1} \cdot 2^{2 n - 1})}^{2}}$

Step 1.4.2

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

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Step 1.4.2.1

Move $2^{3 n + 3}$ .

$\frac{5^{6 n - 2} \cdot (2^{3 n + 3} \cdot 2^{n})}{{(5^{3 n - 1} \cdot 2^{2 n - 1})}^{2}}$

Step 1.4.2.2

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

$\frac{5^{6 n - 2} \cdot 2^{3 n + 3 + n}}{{(5^{3 n - 1} \cdot 2^{2 n - 1})}^{2}}$

Step 1.4.2.3

Add $3 n$ and $n$ .

$\frac{5^{6 n - 2} \cdot 2^{4 n + 3}}{{(5^{3 n - 1} \cdot 2^{2 n - 1})}^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $5^{3 n - 1} \cdot 2^{2 n - 1}$ .

$\frac{5^{6 n - 2} \cdot 2^{4 n + 3}}{{(5^{3 n - 1})}^{2} \cdot {(2^{2 n - 1})}^{2}}$

Step 2.2

Multiply the exponents in ${(5^{3 n - 1})}^{2}$ .

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Step 2.2.1

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

$\frac{5^{6 n - 2} \cdot 2^{4 n + 3}}{5^{(3 n - 1) \cdot 2} \cdot {(2^{2 n - 1})}^{2}}$

Step 2.2.2

Apply the distributive property.

$\frac{5^{6 n - 2} \cdot 2^{4 n + 3}}{5^{3 n \cdot 2 - 1 \cdot 2} \cdot {(2^{2 n - 1})}^{2}}$

Step 2.2.3

Multiply $2$ by $3$ .

$\frac{5^{6 n - 2} \cdot 2^{4 n + 3}}{5^{6 n - 1 \cdot 2} \cdot {(2^{2 n - 1})}^{2}}$

Step 2.2.4

Multiply $- 1$ by $2$ .

$\frac{5^{6 n - 2} \cdot 2^{4 n + 3}}{5^{6 n - 2} \cdot {(2^{2 n - 1})}^{2}}$

Step 2.3

Multiply the exponents in ${(2^{2 n - 1})}^{2}$ .

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Step 2.3.1

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

$\frac{5^{6 n - 2} \cdot 2^{4 n + 3}}{5^{6 n - 2} \cdot 2^{(2 n - 1) \cdot 2}}$

Step 2.3.2

Apply the distributive property.

$\frac{5^{6 n - 2} \cdot 2^{4 n + 3}}{5^{6 n - 2} \cdot 2^{2 n \cdot 2 - 1 \cdot 2}}$

Step 2.3.3

Multiply $2$ by $2$ .

$\frac{5^{6 n - 2} \cdot 2^{4 n + 3}}{5^{6 n - 2} \cdot 2^{4 n - 1 \cdot 2}}$

Step 2.3.4

Multiply $- 1$ by $2$ .

$\frac{5^{6 n - 2} \cdot 2^{4 n + 3}}{5^{6 n - 2} \cdot 2^{4 n - 2}}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $5^{6 n - 2}$ .

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Step 3.1.1

Cancel the common factor.

$\frac{5^{6 n - 2} \cdot 2^{4 n + 3}}{5^{6 n - 2} \cdot 2^{4 n - 2}}$

Step 3.1.2

Rewrite the expression.

$\frac{2^{4 n + 3}}{2^{4 n - 2}}$

Step 3.2

Cancel the common factor of $2^{4 n + 3}$ and $2^{4 n - 2}$ .

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Step 3.2.1

Factor $2^{4 n - 2}$ out of $2^{4 n + 3}$ .

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Multiply by $1$ .

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.2.2.4

Divide $2^{4 n + 3 - (4 n - 2)}$ by $1$ .

$2^{4 n + 3 - (4 n - 2)}$

Step 3.3

Simplify each term.

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Step 3.3.1

Apply the distributive property.

$2^{4 n + 3 - (4 n) - - 2}$

Step 3.3.2

Multiply $4$ by $- 1$ .

$2^{4 n + 3 - 4 n - - 2}$

Step 3.3.3

Multiply $- 1$ by $- 2$ .

$2^{4 n + 3 - 4 n + 2}$

Step 3.4

Simplify by adding terms.

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Step 3.4.1

Combine the opposite terms in $4 n + 3 - 4 n + 2$ .

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Step 3.4.1.1

Subtract $4 n$ from $4 n$ .

$2^{0 + 3 + 2}$

Step 3.4.1.2

Add $0$ and $3$ .

$2^{3 + 2}$

Step 3.4.2

Simplify the expression.

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Step 3.4.2.1

Add $3$ and $2$ .

$2^{5}$

Step 3.4.2.2

Raise $2$ to the power of $5$ .

$32$