If {\left( {\root 3 \of 4 } \right)^{2x + \frac{1}{2}}} = \frac{1}{{32}}, then x =

To solve the equation

\[
\left( \sqrt[3]{4} \right)^{2x + \frac{1}{2}} = \frac{1}{32},
\]

we can start by expressing everything with the same base. Notice that \( \sqrt[3]{4} = 4^{1/3} \) and \( 4 = 2^2 \), so:

\[
4^{1/3} = (2^2)^{1/3} = 2^{2/3}.
\]

Therefore, the left-hand side of the equation becomes:

\[
\left( 2^{2/3} \right)^{2x + \frac{1}{2}} = 2^{\frac{2}{3} \cdot (2x + \frac{1}{2})}.
\]

Now simplify the exponent:

\[
2^{\frac{2}{3} \cdot (2x + \frac{1}{2})} = 2^{\frac{4x}{3} + \frac{1}{3}}.
\]

The right-hand side of the equation is \( \frac{1}{32} \). Since \( 32 = 2^5 \), we can write it as:

\[
\frac{1}{32} = 2^{-5}.
\]

Now we equate the exponents of 2 from both sides of the equation:

\[
\frac{4x}{3} + \frac{1}{3} = -5.
\]

To solve for \( x \), multiply every term by 3 to eliminate the fractions:

\[
4x + 1 = -15.
\]

Now, solve for \( x \) by isolating it:

\[
4x = -15 - 1,
\]

\[
4x = -16,
\]

\[
x = -4.
\]

Thus, the solution is:

\[
\boxed{x = -4}.
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do you express roots as exponents?
2. What are the properties of exponents?
3. How can we handle equations with different bases?
4. What are the steps to simplify equations involving fractional exponents?
5. How can you check the solution to an exponential equation?

**Tip:** Always try to express numbers with a common base when solving exponential equations.

Learn how to solve the exponential equation involving fractional exponents step-by-step. This problem covers key concepts such as exponential equations, fractional exponents, and properties of exponents. Suitable for high school students.

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