Questions: Which of the following is a like radical to ∛(6x²)? x(∛(6x)) 6(∛(x²)) 4(∛(6x²)) x(∛(6))

Solution Steps

To determine which of the given expressions is a like radical to \(\sqrt[3]{6 x^{2}}\), we need to identify the expression that has the same radicand (the expression inside the radical) and the same index (the root).

1. Identify the radicand and the index of the given radical \(\sqrt[3]{6 x^{2}}\).
2. Compare the radicand and the index with those of the given options.
3. The expression with the same radicand and index is the like radical.

The given radical is \( \sqrt[3]{6 x^{2}} \). We can express this as: \[ \sqrt[3]{6 x^{2}} = 6^{\frac{1}{3}} (x^{2})^{\frac{1}{3}} = 6^{\frac{1}{3}} x^{\frac{2}{3}} \]

We will analyze each option to see if it has the same radicand and index as the given radical.

1. Option 1: \( x \sqrt[3]{6 x} = x \cdot 6^{\frac{1}{3}} x^{\frac{1}{3}} = 6^{\frac{1}{3}} x^{\frac{4}{3}} \)
2. Option 2: \( 6 \sqrt[3]{x^{2}} = 6 \cdot (x^{2})^{\frac{1}{3}} = 6^{1} x^{\frac{2}{3}} \)
3. Option 3: \( 4 \sqrt[3]{6 x^{2}} = 4 \cdot \sqrt[3]{6 x^{2}} = 4 \cdot 6^{\frac{1}{3}} x^{\frac{2}{3}} \)
4. Option 4: \( x \sqrt[3]{6} = x \cdot 6^{\frac{1}{3}} = 6^{\frac{1}{3}} x^{1} \)

Now we compare each option with \( 6^{\frac{1}{3}} x^{\frac{2}{3}} \):

• Option 1: \( 6^{\frac{1}{3}} x^{\frac{4}{3}} \) (not a like radical)
• Option 2: \( 6^{1} x^{\frac{2}{3}} \) (not a like radical)
• Option 3: \( 4 \cdot 6^{\frac{1}{3}} x^{\frac{2}{3}} \) (like radical)
• Option 4: \( 6^{\frac{1}{3}} x^{1} \) (not a like radical)

Final Answer