Question

Which expression simplifies to 8 * root(7, 3) ? (1 point) 2 * root(56, 3) - 4 * root(7, 3); - 6 * root(7, 3) - 2 * root(7, 3); root(56, 3) + 6 * root(7, 3); 3 * root(6, 3) + 5 * root(1, 3)

Answer 1

To simplify the given options and find which one equals \( 8 \cdot \sqrt[3]{7} \), we'll simplify each expression step by step.

1. **Option 1:** \( 2 \cdot \sqrt[3]{56} - 4 \cdot \sqrt[3]{7} \)

First, note that \( 56 = 8 \cdot 7 \), so:
\[
\sqrt[3]{56} = \sqrt[3]{8 \cdot 7} = \sqrt[3]{8} \cdot \sqrt[3]{7} = 2 \cdot \sqrt[3]{7}
\]
Thus:
\[
2 \cdot \sqrt[3]{56} = 2 \cdot (2 \cdot \sqrt[3]{7}) = 4 \cdot \sqrt[3]{7}
\]
So, the expression becomes:
\[
4 \cdot \sqrt[3]{7} - 4 \cdot \sqrt[3]{7} = 0
\]

2. **Option 2:** \( -6 \cdot \sqrt[3]{7} - 2 \cdot \sqrt[3]{7} \)

Combine the terms:
\[
-6 \cdot \sqrt[3]{7} - 2 \cdot \sqrt[3]{7} = -8 \cdot \sqrt[3]{7}
\]

3. **Option 3:** \( \sqrt[3]{56} + 6 \cdot \sqrt[3]{7} \)

Again, using \( \sqrt[3]{56} = 2 \cdot \sqrt[3]{7} \), we have:
\[
\sqrt[3]{56} + 6 \cdot \sqrt[3]{7} = 2 \cdot \sqrt[3]{7} + 6 \cdot \sqrt[3]{7} = 8 \cdot \sqrt[3]{7}
\]

4. **Option 4:** \( 3 \cdot \sqrt[3]{6} + 5 \cdot \sqrt[3]{1} \)

Since \( \sqrt[3]{1} = 1 \), this simplifies to:
\[
3 \cdot \sqrt[3]{6} + 5
\]
This does not match \( 8 \cdot \sqrt[3]{7} \).

The only option that simplifies to \( 8 \cdot \sqrt[3]{7} \) is **Option 3**: \( \sqrt[3]{56} + 6 \cdot \sqrt[3]{7} \).