Solve (sqrt{125}+sqrt{5}):sqrt[4]{25}=6

$$( \sqrt { 125 } + \sqrt { 5 } ) : \sqrt[ 4 ] { 25 } = 6$$

Verify

Solution Steps

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

$$\frac{5\sqrt{5}+\sqrt{5}}{\sqrt[4]{25}}=6$$

Combine 5\sqrt{5} and \sqrt{5} to get 6\sqrt{5}.

$$\frac{6\sqrt{5}}{\sqrt[4]{25}}=6$$

Rewrite \sqrt[4]{25} as \sqrt[4]{5^{2}}. Convert from radical to exponential form and cancel out 2 in the exponent. Convert back to radical form.

$$\sqrt[4]{25}=\sqrt[4]{5^{2}}=5^{\frac{2}{4}}=5^{\frac{1}{2}}=\sqrt{5}$$

Insert the obtained value back in the expression.

$$\frac{6\sqrt{5}}{\sqrt{5}}=6$$

Rationalize the denominator of \frac{6\sqrt{5}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{6\sqrt{5}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}=6$$

The square of \sqrt{5} is 5.

$$\frac{6\sqrt{5}\sqrt{5}}{5}=6$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{6\times 5}{5}=6$$

Multiply 6 and 5 to get 30.

$$\frac{30}{5}=6$$

Divide 30 by 5 to get 6.

$$6=6$$

Compare 6 and 6.

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