Basic Math Examples

$\frac{54^{\frac{2}{3}}}{\sqrt{45}} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

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 $45$ .

$\frac{54^{\frac{2}{3}}}{\sqrt{9 (5)}} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 1.1.2

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

$\frac{54^{\frac{2}{3}}}{\sqrt{3^{2} \cdot 5}} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 1.2

Pull terms out from under the radical.

$\frac{54^{\frac{2}{3}}}{3 \sqrt{5}} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 2

Multiply $\frac{54^{\frac{2}{3}}}{3 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{54^{\frac{2}{3}}}{3 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{54^{\frac{2}{3}}}{3 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{54^{\frac{2}{3}} \sqrt{5}}{3 \sqrt{5} \sqrt{5}} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 3.2

Move $\sqrt{5}$ .

$\frac{54^{\frac{2}{3}} \sqrt{5}}{3 (\sqrt{5} \sqrt{5})} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 3.3

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

$\frac{54^{\frac{2}{3}} \sqrt{5}}{3 ({\sqrt{5}}^{1} \sqrt{5})} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 3.4

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

$\frac{54^{\frac{2}{3}} \sqrt{5}}{3 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 3.5

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

$\frac{54^{\frac{2}{3}} \sqrt{5}}{3 {\sqrt{5}}^{1 + 1}} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 3.6

Add $1$ and $1$ .

$\frac{54^{\frac{2}{3}} \sqrt{5}}{3 {\sqrt{5}}^{2}} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 3.7

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

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Step 3.7.1

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

$\frac{54^{\frac{2}{3}} \sqrt{5}}{3 {(5^{\frac{1}{2}})}^{2}} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 3.7.2

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

$\frac{54^{\frac{2}{3}} \sqrt{5}}{3 \cdot 5^{\frac{1}{2} \cdot 2}} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 3.7.3

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

$\frac{54^{\frac{2}{3}} \sqrt{5}}{3 \cdot 5^{\frac{2}{2}}} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

$\frac{54^{\frac{2}{3}} \sqrt{5}}{3 \cdot 5^{\frac{2}{2}}} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 3.7.4.2

Rewrite the expression.

$\frac{54^{\frac{2}{3}} \sqrt{5}}{3 \cdot 5^{1}} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 3.7.5

Evaluate the exponent.

$\frac{54^{\frac{2}{3}} \sqrt{5}}{3 \cdot 5} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 4

Multiply $3$ by $5$ .

$\frac{54^{\frac{2}{3}} \sqrt{5}}{15} \cdot {(\frac{125}{4})}^{- \frac{1}{6}}$

Step 5

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{54^{\frac{2}{3}} \sqrt{5}}{15} \cdot {(\frac{4}{125})}^{\frac{1}{6}}$

Step 6

Apply the product rule to $\frac{4}{125}$ .

$\frac{54^{\frac{2}{3}} \sqrt{5}}{15} \cdot \frac{4^{\frac{1}{6}}}{125^{\frac{1}{6}}}$

Step 7

Combine.

$\frac{54^{\frac{2}{3}} \sqrt{5} \cdot 4^{\frac{1}{6}}}{15 \cdot 125^{\frac{1}{6}}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{54^{\frac{2}{3}} \sqrt{5} \cdot 4^{\frac{1}{6}}}{15 \cdot 125^{\frac{1}{6}}}$

Decimal Form:

$1.2$