2) $$ \sqrt[a]{32 x^{2}}=2 x^{\frac{2}{b}} $$

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UpStudy AI · Accepted Answer

To solve the equation:

$$
\sqrt[a]{32 x^{2}} = 2 x^{\frac{2}{b}}
$$

we need to find the values of $$ a $$ and $$ b $$ that make this equation true for all $$ x $$. Let's break it down step by step.

### Step 1: Rewrite the Radical as an Exponent

Recall that the $$ a $$-th root of a number is the same as raising that number to the power of $$ \frac{1}{a} $$. Thus, we can rewrite the left side of the equation:

$$
\sqrt[a]{32 x^{2}} = (32 x^{2})^{\frac{1}{a}}
$$

### Step 2: Simplify the Expression

Apply the exponent to both the constant and the variable:

$$
(32 x^{2})^{\frac{1}{a}} = 32^{\frac{1}{a}} \cdot x^{\frac{2}{a}}
$$

So now the equation looks like:

$$
32^{\frac{1}{a}} \cdot x^{\frac{2}{a}} = 2 \cdot x^{\frac{2}{b}}
$$

### Step 3: Equate the Coefficients and the Exponents

For the equation to hold true for all $$ x $$, both the coefficients and the exponents must be equal on both sides.

**a. Equate the Coefficients:**

$$
32^{\frac{1}{a}} = 2
$$

Recall that $$ 32 = 2^5 $$, so:

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

Since the bases are the same, set the exponents equal:

$$
\frac{5}{a} = 1 \
a = 5
$$

**b. Equate the Exponents:**

$$
\frac{2}{a} = \frac{2}{b}
$$

Substitute $$ a = 5 $$:

$$
\frac{2}{5} = \frac{2}{b}
$$

Cross-multiply to solve for $$ b $$:

$$
2b = 10 \
b = 5
$$

### Conclusion

The values of $$ a $$ and $$ b $$ that satisfy the equation are:

$$
a = 5 \
b = 5
$$
$$ a = 5 $$ and $$ b = 5 $$.

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