Pre-Algebra Examples

$\frac{9^{n + 1}}{27^{n}} \div \frac{24^{3}}{3^{n + 2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{9^{n + 1}}{27^{n}} \cdot \frac{3^{n + 2}}{24^{3}}$

Step 2

Combine.

$\frac{9^{n + 1} \cdot 3^{n + 2}}{27^{n} \cdot 24^{3}}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $9$ as $3^{2}$ .

$\frac{{(3^{2})}^{n + 1} \cdot 3^{n + 2}}{27^{n} \cdot 24^{3}}$

Step 3.2

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

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Step 3.2.1

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

$\frac{3^{2 (n + 1)} \cdot 3^{n + 2}}{27^{n} \cdot 24^{3}}$

Step 3.2.2

Apply the distributive property.

$\frac{3^{2 n + 2 \cdot 1} \cdot 3^{n + 2}}{27^{n} \cdot 24^{3}}$

Step 3.2.3

Multiply $2$ by $1$ .

$\frac{3^{2 n + 2} \cdot 3^{n + 2}}{27^{n} \cdot 24^{3}}$

Step 3.3

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

$\frac{3^{2 n + 2 + n + 2}}{27^{n} \cdot 24^{3}}$

Step 3.4

Add $2 n$ and $n$ .

$\frac{3^{3 n + 2 + 2}}{27^{n} \cdot 24^{3}}$

Step 3.5

Add $2$ and $2$ .

$\frac{3^{3 n + 4}}{27^{n} \cdot 24^{3}}$

Step 4

Raise $24$ to the power of $3$ .

$\frac{3^{3 n + 4}}{27^{n} \cdot 13824}$

Step 5

Move $13824$ to the left of $27^{n}$ .

$\frac{3^{3 n + 4}}{13824 \cdot 27^{n}}$