`root3(3-17/27)=`

Options

A. `4/3`
B. `3/4`
C. `1/4`
D. `1/2`

Answer

To solve the expression \( \sqrt[3]{3 - \frac{17}{27}} \), we will follow these steps:

### Step 1: Simplify the expression inside the cube root
We start with the expression:
\[ 3 - \frac{17}{27} \]
To combine these, we need a common denominator. The common denominator for 3 (which is \( \frac{3 \times 27}{27} = \frac{81}{27} \)) and \( \frac{17}{27} \) is 27.

### Step 2: Rewrite 3 with the common denominator
Now we can rewrite 3:
\[ 3 = \frac{81}{27} \]
So, we can rewrite the expression as:
\[ \frac{81}{27} - \frac{17}{27} = \frac{81 - 17}{27} \]

### Step 3: Perform the subtraction
Now we perform the subtraction in the numerator:
\[ 81 - 17 = 64 \]
So, we have:
\[ \frac{64}{27} \]

### Step 4: Substitute back into the cube root
Now we substitute this back into the cube root:
\[ \sqrt[3]{\frac{64}{27}} \]

### Step 5: Simplify the cube root
We can separate the cube root of the fraction:
\[ \sqrt[3]{\frac{64}{27}} = \frac{\sqrt[3]{64}}{\sqrt[3]{27}} \]
We know that:
\[ \sqrt[3]{64} = 4 \quad \text{(since \( 4^3 = 64 \))} \]
and
\[ \sqrt[3]{27} = 3 \quad \text{(since \( 3^3 = 27 \))} \]

### Step 6: Write the final answer
Thus, we have:
\[ \frac{\sqrt[3]{64}}{\sqrt[3]{27}} = \frac{4}{3} \]

So, the final answer is:
\[ \sqrt[3]{3 - \frac{17}{27}} = \frac{4}{3} \]