Basic Math Examples

$\frac{42}{\sqrt{29} \cdot \sqrt{80}}$

Step 1

Simplify the denominator.

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Step 1.1

Combine using the product rule for radicals.

$\frac{42}{\sqrt{29 \cdot 80}}$

Step 1.2

Multiply $29$ by $80$ .

$\frac{42}{\sqrt{2320}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $2320$ as $4^{2} \cdot 145$ .

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Step 2.1.1

Factor $16$ out of $2320$ .

$\frac{42}{\sqrt{16 (145)}}$

Step 2.1.2

Rewrite $16$ as $4^{2}$ .

$\frac{42}{\sqrt{4^{2} \cdot 145}}$

Step 2.2

Pull terms out from under the radical.

$\frac{42}{4 \sqrt{145}}$

Step 3

Cancel the common factor of $42$ and $4$ .

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Step 3.1

Factor $2$ out of $42$ .

$\frac{2 (21)}{4 \sqrt{145}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $2$ out of $4 \sqrt{145}$ .

$\frac{2 (21)}{2 (2 \sqrt{145})}$

Step 3.2.2

Cancel the common factor.

$\frac{2 \cdot 21}{2 (2 \sqrt{145})}$

Step 3.2.3

Rewrite the expression.

$\frac{21}{2 \sqrt{145}}$

Step 4

Multiply $\frac{21}{2 \sqrt{145}}$ by $\frac{\sqrt{145}}{\sqrt{145}}$ .

$\frac{21}{2 \sqrt{145}} \cdot \frac{\sqrt{145}}{\sqrt{145}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{21}{2 \sqrt{145}}$ by $\frac{\sqrt{145}}{\sqrt{145}}$ .

$\frac{21 \sqrt{145}}{2 \sqrt{145} \sqrt{145}}$

Step 5.2

Move $\sqrt{145}$ .

$\frac{21 \sqrt{145}}{2 (\sqrt{145} \sqrt{145})}$

Step 5.3

Raise $\sqrt{145}$ to the power of $1$ .

$\frac{21 \sqrt{145}}{2 ({\sqrt{145}}^{1} \sqrt{145})}$

Step 5.4

Raise $\sqrt{145}$ to the power of $1$ .

$\frac{21 \sqrt{145}}{2 ({\sqrt{145}}^{1} {\sqrt{145}}^{1})}$

Step 5.5

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

$\frac{21 \sqrt{145}}{2 {\sqrt{145}}^{1 + 1}}$

Step 5.6

Add $1$ and $1$ .

$\frac{21 \sqrt{145}}{2 {\sqrt{145}}^{2}}$

Step 5.7

Rewrite ${\sqrt{145}}^{2}$ as $145$ .

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Step 5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{145}$ as $145^{\frac{1}{2}}$ .

$\frac{21 \sqrt{145}}{2 {(145^{\frac{1}{2}})}^{2}}$

Step 5.7.2

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

$\frac{21 \sqrt{145}}{2 \cdot 145^{\frac{1}{2} \cdot 2}}$

Step 5.7.3

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

$\frac{21 \sqrt{145}}{2 \cdot 145^{\frac{2}{2}}}$

Step 5.7.4

Cancel the common factor of $2$ .

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Step 5.7.4.1

Cancel the common factor.

$\frac{21 \sqrt{145}}{2 \cdot 145^{\frac{2}{2}}}$

Step 5.7.4.2

Rewrite the expression.

$\frac{21 \sqrt{145}}{2 \cdot 145^{1}}$

Step 5.7.5

Evaluate the exponent.

$\frac{21 \sqrt{145}}{2 \cdot 145}$

Step 6

Multiply $2$ by $145$ .

$\frac{21 \sqrt{145}}{290}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{21 \sqrt{145}}{290}$

Decimal Form:

$0.87197753 \dots$