Question

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 \( x^{1/4} \).