Question

Simplify the expression below. Show all steps and calculations to earn full credit. You may want to do this work by hand and upload an image of that written work rather than try to type it all out. \sqrt[]{\frac{ \sqrt[3]{64}+\sqrt[4]{256}}{\sqrt[]{64}+\sqrt[]{256}}}

Answer 1

The solution and the steps to the simplification of the given surd expression is
`\frac{√( 3) }{{ 3} }`

Simplification of surds.

Surds are mathematical expression that cannot be simplify by ordinary ways of dividing the denominator but its simplification follows some rules.

Given that:

`\sqrt{ \frac{\sqrt[3]{64} + \sqrt[4]{256}} {√(64)+ √(256)}}`

We will first find the square root and the cube root of each parameter in the whole entity as the case maybe;

`\sqrt{ \frac{4 + 4} {8+16} }`

`\sqrt{ \frac{8} {24} }`

Square the numerator and denominator separately.

`\frac{√(8)} {√(24)} }`

Expand each parameter for further simplification

`\frac{√(4 * 2)} {√(4* 6)} }`

`(2√(2))/( 2 √(6)) }`

To expand and solve further.

`(√(2))/( √(6)) }`

`\frac{√(2) * √(6) }{{ √(6)} * √(6)} }`

`\frac{√(12) }{{ 6} }`

`\frac{√(4* 3) }{{ 6} }`

`\frac{2√( 3) }{{ 6} }`

`\frac{√( 3) }{{ 3} }`

Answer 2

We are given the expression:

`\sqrt[]{\frac{ \sqrt[3]{64}+\sqrt[4]{256}}{\sqrt[]{64}+\sqrt[]{256}}}`

We will simplify this as shown below:

`\begin{gathered} \sqrt[]{\frac{ \sqrt[3]{64}+\sqrt[4]{256}}{\sqrt[]{64}+\sqrt[]{256}}} \\ \text{Let's consider \& solve the terms one after the order, we have:} \\ \sqrt[3]{64}\Rightarrow\sqrt[3]{4*4*4}\Rightarrow4 \\ \sqrt[4]{256}\Rightarrow\sqrt[4]{4*4*4*4}\Rightarrow4 \\ \sqrt[]{64}\Rightarrow\sqrt[]{8*8}\Rightarrow8 \\ \sqrt[]{256}\Rightarrow\sqrt[]{16*16}\Rightarrow16 \\ We\text{ will substitute each of these expressions back into the parent expression, we have:} \\ \sqrt[]{(4+4)/(8+16)} \\ =\sqrt[]{(8)/(24)} \\ =\sqrt[]{(1)/(3)} \\ =\frac{\sqrt[]{3}}{\sqrt[]{3}*\sqrt[]{3}} \\ =\frac{\sqrt[]{3}}{3} \\ \Rightarrow\sqrt[]{\frac{\sqrt[3]{64}+\sqrt[4]{256}}{\sqrt[]{64}+\sqrt[]{256}}}=\frac{\sqrt[]{3}}{3} \\ \\ \therefore\frac{\sqrt[]{3}}{3} \end{gathered}`