Algebra Examples

$\sqrt{\frac{23}{17}}$

Step 1

Rewrite $\sqrt{\frac{23}{17}}$ as $\frac{\sqrt{23}}{\sqrt{17}}$ .

$\frac{\sqrt{23}}{\sqrt{17}}$

Step 2

Multiply to rationalize the numerator.

$\frac{\sqrt{23} \sqrt{23}}{\sqrt{17} \sqrt{23}}$

Step 3

Simplify.

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Step 3.1

Raise $\sqrt{23}$ to the power of $1$ .

$\frac{{\sqrt{23}}^{1} \sqrt{23}}{\sqrt{17} \sqrt{23}}$

Step 3.2

Raise $\sqrt{23}$ to the power of $1$ .

$\frac{{\sqrt{23}}^{1} {\sqrt{23}}^{1}}{\sqrt{17} \sqrt{23}}$

Step 3.3

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

$\frac{{\sqrt{23}}^{1 + 1}}{\sqrt{17} \sqrt{23}}$

Step 3.4

Add $1$ and $1$ .

$\frac{{\sqrt{23}}^{2}}{\sqrt{17} \sqrt{23}}$

Step 3.5

Rewrite ${\sqrt{23}}^{2}$ as $23$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{23}$ as $23^{\frac{1}{2}}$ .

$\frac{{(23^{\frac{1}{2}})}^{2}}{\sqrt{17} \sqrt{23}}$

Step 3.5.2

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

$\frac{23^{\frac{1}{2} \cdot 2}}{\sqrt{17} \sqrt{23}}$

Step 3.5.3

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

$\frac{23^{\frac{2}{2}}}{\sqrt{17} \sqrt{23}}$

Step 3.5.4

Cancel the common factor of $2$ .

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Step 3.5.4.1

Cancel the common factor.

$\frac{23^{\frac{2}{2}}}{\sqrt{17} \sqrt{23}}$

Step 3.5.4.2

Rewrite the expression.

$\frac{23^{1}}{\sqrt{17} \sqrt{23}}$

Step 3.5.5

Evaluate the exponent.

$\frac{23}{\sqrt{17} \sqrt{23}}$

Step 3.6

Combine using the product rule for radicals.

$\frac{23}{\sqrt{17 \cdot 23}}$

Step 3.7

Multiply $17$ by $23$ .

$\frac{23}{\sqrt{391}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{23}{\sqrt{391}}$

Decimal Form:

$1.16315999 \dots$