Evaluate the left hand side to find the value of $$ a $$ in the equation is simplest form. $$ \left(x^{\frac{1}{2}}\right)^{\frac{6}{5}}=x^{a} $$

Question

UpStudy AI · Accepted Answer

To find the value of $$ a $$ in the equation:

$$
\left(x^{\frac{1}{2}}\right)^{\frac{6}{5}} = x^{a}
$$

we can simplify the left-hand side using the **power of a power rule**, which states that $$(x^m)^n = x^{m \cdot n}$$.

**Step-by-Step Simplification:**

1. **Apply the Power of a Power Rule:**

$$
\left(x^{\frac{1}{2}}\right)^{\frac{6}{5}} = x^{\frac{1}{2} \times \frac{6}{5}}
$$

2. **Multiply the Exponents:**

$$
\frac{1}{2} \times \frac{6}{5} = \frac{6}{10} = \frac{3}{5}
$$

3. **Rewrite the Expression:**

$$
x^{\frac{1}{2} \times \frac{6}{5}} = x^{\frac{3}{5}}
$$

So, the equation becomes:

$$
x^{\frac{3}{5}} = x^{a}
$$

Since the bases are the same and the expressions are equal, the exponents must be equal:

$$
a = \frac{3}{5}
$$

**Final Answer:**

$$
a = \frac{3}{5}
$$
$$ a = \frac{3}{5} $$

Evaluate the left hand side to find the value of \( a \) in the equation is simplest form. \[ \left(x^{\frac{1}{2}}\right)^{\frac{6}{5}}=x^{a} \]