1. **State the problem:** We are given the equation $32\sqrt{2} = 2x^a$ and need to find the value of $a$. 2. **Rewrite the equation:** Note that $32 = 2^5$ and $\sqrt{2} = 2^{\frac{1}{2}}$. So, $$32\sqrt{2} = 2^5 \times 2^{\frac{1}{2}} = 2^{5 + \frac{1}{2}} = 2^{\frac{11}{2}}.$$ 3. **Express the right side:** The right side is $2x^a$. To compare powers of 2, express $x$ as a power of 2, say $x = 2^k$. Then, $$2x^a = 2 \times (2^k)^a = 2 \times 2^{ka} = 2^{1 + ka}.$$ 4. **Equate the exponents:** Since the bases are the same (base 2), equate the exponents: $$\frac{11}{2} = 1 + ka.$$ 5. **Solve for $a$:** $$ka = \frac{11}{2} - 1 = \frac{9}{2}.$$ 6. **Conclusion:** Without knowing $k$ (the exponent in $x = 2^k$), we cannot find a unique value for $a$. If $x$ is known, then $$a = \frac{9}{2k}.$$ **Summary:** The value of $a$ depends on $x$. If $x$ is a power of 2, say $x = 2^k$, then $$a = \frac{9}{2k}.$$