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

Question

UpStudy AI · Accepted Answer

To solve the equation:

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

let's break it down step by step.

### Step 1: Rewrite the Radical as an Exponent
The radical $$\sqrt[a]{32 x^{2}}$$ can be rewritten using fractional exponents:

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

### Step 2: Express Constants as Powers of 2
Notice that $$32$$ is a power of $$2$$:

$$
32 = 2^5
$$

So, substitute this into the equation:

$$
(2^5 x^2)^{1/a} = 2 x^{\frac{2}{b}}
$$

### Step 3: Apply the Exponent to Both Terms Inside the Parenthesis
Using the property $$(ab)^c = a^c b^c$$:

$$
2^{5/a} \cdot x^{2/a} = 2 \cdot x^{2/b}
$$

### Step 4: Equate the Exponents of Like Bases
Since the bases on both sides of the equation must be equal, their exponents must also be equal.

**For the base 2:**

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

**For the base $$x$$:**

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

Substitute $$a = 5$$:

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

### Conclusion
The values of $$a$$ and $$b$$ that satisfy the equation are both $$5$$.

$$
\boxed{a = 5,\ b = 5}
$$
$$ a = 5 $$ and $$ b = 5 $$.

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