Question

simplify the expression: cube root of 96x to the power of 7

2x to the power of 2 cube root of 32x
32x cube root of 12x to the power of 4
9x cube root of 12x to the power of 2
2x to the power of 2 cube root of 12x

Answer 1

To simplify the expression \( \sqrt[3]{96x^7} \), we will break it down into manageable parts.

1. **Factor the coefficient and the variable:**
- Start with \( 96 \):
\[
96 = 2^5 \times 3
\]
- For the variable \( x^7 \), we can write:
\[
x^7 = (x^6) \times x = (x^2)^3 \times x
\]

2. **Combine everything under the cube root:**
\[
\sqrt[3]{96x^7} = \sqrt[3]{(2^5 \times 3) \times (x^2)^3 \times x}
\]

3. **Use the property of cube roots**:
\[
\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}
\]
Thus, we can separate it:
\[
\sqrt[3]{96x^7} = \sqrt[3]{2^5} \times \sqrt[3]{3} \times \sqrt[3]{(x^2)^3} \times \sqrt[3]{x}
\]

4. **Calculate cubes:**
- \( \sqrt[3]{(x^2)^3} = x^2 \)
- \( \sqrt[3]{2^5} = \sqrt[3]{2^3 \cdot 2^2} = 2 \sqrt[3]{4} \) (since \( \sqrt[3]{2^3} = 2 \))

5. **Putting it all together**:
\[
\sqrt[3]{96x^7} = 2 \sqrt[3]{4} \times \sqrt[3]{3} \times x^2 \times \sqrt[3]{x}
\]
\[
= 2 x^2 \sqrt[3]{12x^4}
\]

Thus, the simplified expression is:
\[
2x^2 \sqrt[3]{12x^4}
\]

Since \( \sqrt[3]{12} = \sqrt[3]{4 \cdot 3} = \sqrt[3]{4} \cdot \sqrt[3]{3} \), you could keep it in this form or express it as a mixed term, but \( 2x^2 \sqrt[3]{12x^4} \) is the most straightforward simplified expression.