Using the properties of exponents which expression is equivalent to x^1/6 times x ^1/6 square rooted

Answer 1

To simplify the expression \( x^{1/6} \times \sqrt{x^{1/6}} \), we can follow these steps:

1. Rewrite the square root: The square root of any expression can be rewritten in terms of exponents. We know that: \[ \sqrt{a} = a^{1/2} \] So, we can write: \[ \sqrt{x^{1/6}} = (x^{1/6})^{1/2} \]

2. Multiply the exponents: When raising a power to another power, you multiply the exponents: \[ (x^{1/6})^{1/2} = x^{(1/6) \cdot (1/2)} = x^{1/12} \]

3. Combine the expressions: Now substitute this back into the original expression: \[ x^{1/6} \times x^{1/12} \]

4. Use the properties of exponents: When multiplying two powers with the same base, you add the exponents: \[ x^{1/6} \times x^{1/12} = x^{(1/6) + (1/12)} \]

5. Find a common denominator: The least common multiple of 6 and 12 is 12: \[ \frac{1}{6} = \frac{2}{12} \] So: \[ (1/6) + (1/12) = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \]

6. Final expression: Therefore, we have: \[ x^{1/6} \times \sqrt{x^{1/6}} = x^{1/4} \]

So, the expression equivalent to \( x^{1/6} \times \sqrt{x^{1/6}} \) is \( x^{1/4} \).