Algebra Examples

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

Step 1

Move $3^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Raise $2$ to the power of $3$ .

$\frac{8 \cdot {(3^{- 2})}^{3} \cdot 5^{2} \cdot {(3^{2})}^{2} \cdot 3^{2}}{{(5 \cdot 2)}^{2}}$

Step 2.4

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

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Step 2.4.1

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

$\frac{8 \cdot 3^{- 2 \cdot 3} \cdot 5^{2} \cdot {(3^{2})}^{2} \cdot 3^{2}}{{(5 \cdot 2)}^{2}}$

Step 2.4.2

Multiply $- 2$ by $3$ .

$\frac{8 \cdot 3^{- 6} \cdot 5^{2} \cdot {(3^{2})}^{2} \cdot 3^{2}}{{(5 \cdot 2)}^{2}}$

Step 2.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{8 \cdot \frac{1}{3^{6}} \cdot 5^{2} \cdot {(3^{2})}^{2} \cdot 3^{2}}{{(5 \cdot 2)}^{2}}$

Step 2.6

Raise $3$ to the power of $6$ .

$\frac{8 \cdot \frac{1}{729} \cdot 5^{2} \cdot {(3^{2})}^{2} \cdot 3^{2}}{{(5 \cdot 2)}^{2}}$

Step 2.7

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

$\frac{8 \cdot \frac{1}{729} \cdot 25 \cdot {(3^{2})}^{2} \cdot 3^{2}}{{(5 \cdot 2)}^{2}}$

Step 2.8

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

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Step 2.8.1

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

$\frac{8 \cdot \frac{1}{729} \cdot 25 \cdot 3^{2 \cdot 2} \cdot 3^{2}}{{(5 \cdot 2)}^{2}}$

Step 2.8.2

Multiply $2$ by $2$ .

$\frac{8 \cdot \frac{1}{729} \cdot 25 \cdot 3^{4} \cdot 3^{2}}{{(5 \cdot 2)}^{2}}$

Step 2.9

Raise $3$ to the power of $4$ .

$\frac{8 \cdot \frac{1}{729} \cdot 25 \cdot 81 \cdot 3^{2}}{{(5 \cdot 2)}^{2}}$

Step 2.10

Raise $3$ to the power of $2$ .

$\frac{8 \cdot \frac{1}{729} \cdot 25 \cdot 81 \cdot 9}{{(5 \cdot 2)}^{2}}$

Step 2.11

Combine exponents.

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Step 2.11.1

Combine $8$ and $\frac{1}{729}$ .

$\frac{\frac{8}{729} \cdot 25 \cdot 81 \cdot 9}{{(5 \cdot 2)}^{2}}$

Step 2.11.2

Combine $\frac{8}{729}$ and $25$ .

$\frac{\frac{8 \cdot 25}{729} \cdot 81 \cdot 9}{{(5 \cdot 2)}^{2}}$

Step 2.11.3

Multiply $8$ by $25$ .

$\frac{\frac{200}{729} \cdot 81 \cdot 9}{{(5 \cdot 2)}^{2}}$

Step 2.11.4

Combine $\frac{200}{729}$ and $81$ .

$\frac{\frac{200 \cdot 81}{729} \cdot 9}{{(5 \cdot 2)}^{2}}$

Step 2.11.5

Multiply $200$ by $81$ .

$\frac{\frac{16200}{729} \cdot 9}{{(5 \cdot 2)}^{2}}$

Step 2.11.6

Combine $\frac{16200}{729}$ and $9$ .

$\frac{\frac{16200 \cdot 9}{729}}{{(5 \cdot 2)}^{2}}$

Step 2.11.7

Multiply $16200$ by $9$ .

$\frac{\frac{145800}{729}}{{(5 \cdot 2)}^{2}}$

Step 2.12

Divide $145800$ by $729$ .

$\frac{200}{{(5 \cdot 2)}^{2}}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $5 \cdot 2$ .

$\frac{200}{5^{2} \cdot 2^{2}}$

Step 3.2

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

$\frac{200}{25 \cdot 2^{2}}$

Step 3.3

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

$\frac{200}{25 \cdot 4}$

Step 4

Simplify the expression.

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Step 4.1

Multiply $25$ by $4$ .

$\frac{200}{100}$

Step 4.2

Divide $200$ by $100$ .

$2$