Algebra Examples

$- \frac{9}{2 \sqrt{45}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 1.1.1

Factor $9$ out of $45$ .

$- \frac{9}{2 \sqrt{9 (5)}}$

Step 1.1.2

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

$- \frac{9}{2 \sqrt{3^{2} \cdot 5}}$

Step 1.2

Pull terms out from under the radical.

$- \frac{9}{2 \cdot 3 \sqrt{5}}$

Step 1.3

Multiply $2$ by $3$ .

$- \frac{9}{6 \sqrt{5}}$

Step 2

Cancel the common factor of $9$ and $6$ .

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Step 2.1

Factor $3$ out of $9$ .

$- \frac{3 (3)}{6 \sqrt{5}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $3$ out of $6 \sqrt{5}$ .

$- \frac{3 (3)}{3 (2 \sqrt{5})}$

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

$- \frac{3}{2 \sqrt{5}}$

Step 3

Multiply $\frac{3}{2 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{3}{2 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$- \frac{3 \sqrt{5}}{2 \sqrt{5} \sqrt{5}}$

Step 4.2

Move $\sqrt{5}$ .

$- \frac{3 \sqrt{5}}{2 (\sqrt{5} \sqrt{5})}$

Step 4.3

Raise $\sqrt{5}$ to the power of $1$ .

$- \frac{3 \sqrt{5}}{2 ({\sqrt{5}}^{1} \sqrt{5})}$

Step 4.4

Raise $\sqrt{5}$ to the power of $1$ .

$- \frac{3 \sqrt{5}}{2 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}$

Step 4.5

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

$- \frac{3 \sqrt{5}}{2 {\sqrt{5}}^{1 + 1}}$

Step 4.6

Add $1$ and $1$ .

$- \frac{3 \sqrt{5}}{2 {\sqrt{5}}^{2}}$

Step 4.7

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$- \frac{3 \sqrt{5}}{2 {(5^{\frac{1}{2}})}^{2}}$

Step 4.7.2

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

$- \frac{3 \sqrt{5}}{2 \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 4.7.3

Combine $\frac{1}{2}$ and $2$ .

$- \frac{3 \sqrt{5}}{2 \cdot 5^{\frac{2}{2}}}$

Step 4.7.4

Cancel the common factor of $2$ .

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Step 4.7.4.1

Cancel the common factor.

$- \frac{3 \sqrt{5}}{2 \cdot 5^{\frac{2}{2}}}$

Step 4.7.4.2

Rewrite the expression.

$- \frac{3 \sqrt{5}}{2 \cdot 5^{1}}$

Step 4.7.5

Evaluate the exponent.

$- \frac{3 \sqrt{5}}{2 \cdot 5}$

Step 5

Multiply $2$ by $5$ .

$- \frac{3 \sqrt{5}}{10}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{3 \sqrt{5}}{10}$

Decimal Form:

$- 0.67082039 \dots$