Algebra Examples

$3^{- \frac{1}{2}} \cdot 3 x^{\frac{9}{2}} {(3 x)}^{- \frac{7}{2}}$

Step 1

Multiply $3^{- \frac{1}{2}}$ by $3$ by adding the exponents.

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Step 1.1

Move $3$ .

$3 \cdot 3^{- \frac{1}{2}} \cdot (x^{\frac{9}{2}} {(3 x)}^{- \frac{7}{2}})$

Step 1.2

Multiply $3$ by $3^{- \frac{1}{2}}$ .

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Step 1.2.1

Raise $3$ to the power of $1$ .

$3^{1} \cdot 3^{- \frac{1}{2}} \cdot (x^{\frac{9}{2}} {(3 x)}^{- \frac{7}{2}})$

Step 1.2.2

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

$3^{1 - \frac{1}{2}} \cdot (x^{\frac{9}{2}} {(3 x)}^{- \frac{7}{2}})$

Step 1.3

Write $1$ as a fraction with a common denominator.

$3^{\frac{2}{2} - \frac{1}{2}} \cdot (x^{\frac{9}{2}} {(3 x)}^{- \frac{7}{2}})$

Step 1.4

Combine the numerators over the common denominator.

$3^{\frac{2 - 1}{2}} \cdot (x^{\frac{9}{2}} {(3 x)}^{- \frac{7}{2}})$

Step 1.5

Subtract $1$ from $2$ .

$3^{\frac{1}{2}} \cdot (x^{\frac{9}{2}} {(3 x)}^{- \frac{7}{2}})$

Step 2

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

$3^{\frac{1}{2}} \cdot (x^{\frac{9}{2}} \frac{1}{{(3 x)}^{\frac{7}{2}}})$

Step 3

Simplify terms.

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Step 3.1

Apply the product rule to $3 x$ .

$3^{\frac{1}{2}} \cdot (x^{\frac{9}{2}} \frac{1}{3^{\frac{7}{2}} x^{\frac{7}{2}}})$

Step 3.2

Cancel the common factor of $x^{\frac{7}{2}}$ .

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Step 3.2.1

Factor $x^{\frac{7}{2}}$ out of $x^{\frac{9}{2}}$ .

$3^{\frac{1}{2}} \cdot (x^{\frac{7}{2}} x^{\frac{2}{2}} \frac{1}{3^{\frac{7}{2}} x^{\frac{7}{2}}})$

Step 3.2.2

Factor $x^{\frac{7}{2}}$ out of $3^{\frac{7}{2}} x^{\frac{7}{2}}$ .

$3^{\frac{1}{2}} \cdot (x^{\frac{7}{2}} x^{\frac{2}{2}} \frac{1}{x^{\frac{7}{2}} \cdot 3^{\frac{7}{2}}})$

Step 3.2.3

Cancel the common factor.

$3^{\frac{1}{2}} \cdot (x^{\frac{7}{2}} x^{\frac{2}{2}} \frac{1}{x^{\frac{7}{2}} \cdot 3^{\frac{7}{2}}})$

Step 3.2.4

Rewrite the expression.

$3^{\frac{1}{2}} \cdot (x^{\frac{2}{2}} \frac{1}{3^{\frac{7}{2}}})$

Step 3.3

Combine $x^{\frac{2}{2}}$ and $\frac{1}{3^{\frac{7}{2}}}$ .

$3^{\frac{1}{2}} \cdot \frac{x^{\frac{2}{2}}}{3^{\frac{7}{2}}}$

Step 4

Simplify the numerator.

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Step 4.1

Divide $2$ by $2$ .

$3^{\frac{1}{2}} \cdot \frac{x^{1}}{3^{\frac{7}{2}}}$

Step 4.2

Simplify.

$3^{\frac{1}{2}} \cdot \frac{x}{3^{\frac{7}{2}}}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $3^{\frac{1}{2}}$ .

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Step 5.1.1

Factor $3^{\frac{1}{2}}$ out of $3^{\frac{7}{2}}$ .

$3^{\frac{1}{2}} \cdot \frac{x}{3^{\frac{1}{2}} \cdot 3^{\frac{6}{2}}}$

Step 5.1.2

Cancel the common factor.

$3^{\frac{1}{2}} \cdot \frac{x}{3^{\frac{1}{2}} \cdot 3^{\frac{6}{2}}}$

Step 5.1.3

Rewrite the expression.

$\frac{x}{3^{\frac{6}{2}}}$

Step 5.2

Cancel the common factor of $6$ and $2$ .

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Step 5.2.1

Factor $2$ out of $6$ .

$\frac{x}{3^{\frac{2 \cdot 3}{2}}}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2$ .

$\frac{x}{3^{\frac{2 \cdot 3}{2 (1)}}}$

Step 5.2.2.2

Cancel the common factor.

$\frac{x}{3^{\frac{2 \cdot 3}{2 \cdot 1}}}$

Step 5.2.2.3

Rewrite the expression.

$\frac{x}{3^{\frac{3}{1}}}$

Step 5.2.2.4

Divide $3$ by $1$ .

$\frac{x}{3^{3}}$

Step 5.3

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

$\frac{x}{27}$