Evaluate `root3((27)/(64))`

Answer

To evaluate the expression \( \sqrt[3]{\frac{27}{64}} \), we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the cube root of a fraction as the fraction of the cube roots:
\[ \sqrt[3]{\frac{27}{64}} = \frac{\sqrt[3]{27}}{\sqrt[3]{64}} \]

### Step 2: Calculate the cube root of the numerator
Next, we find the cube root of 27. We know that:
\[ 27 = 3^3 \]
Thus,
\[ \sqrt[3]{27} = 3 \]

### Step 3: Calculate the cube root of the denominator
Now, we find the cube root of 64. We know that:
\[ 64 = 4^3 \]
Thus,
\[ \sqrt[3]{64} = 4 \]

### Step 4: Combine the results
Now we can substitute back into our fraction:
\[ \frac{\sqrt[3]{27}}{\sqrt[3]{64}} = \frac{3}{4} \]

### Final Answer
Therefore, the value of \( \sqrt[3]{\frac{27}{64}} \) is:
\[ \frac{3}{4} \]

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