Evaluating the Expression Inside the Radical

Oct 13, 2025 by leolavanderia.com.br 44 views

Understanding the Problem

The given equation involves a radical expression with a fractional exponent. To simplify and solve for $x$ , we need to carefully evaluate the expression inside the radical and then isolate the variable $x$ . The equation is $\sqrt[\frac{1}{3}]{3\left(\frac{1}{2}+\frac{1}{4}\right)} = \frac{x}{4}$ .

Evaluating the Expression Inside the Radical

The first step is to simplify the expression inside the radical. We can start by finding a common denominator for the fractions $\frac{1}{2}$ and $\frac{1}{4}$ . The common denominator is $4$ , so we can rewrite the fractions as $\frac{2}{4}$ and $\frac{1}{4}$ .

\frac{1}{2}+\frac{1}{4} = \frac{2}{4}+\frac{1}{4} = \frac{3}{4}

Now, we can substitute this simplified expression back into the original equation:

$\sqrt[\frac{1}{3}]{3\left(\frac{3}{4}\right)} = \frac{x}{4}$

Simplifying the Radical Expression

Next, we need to simplify the radical expression. We can start by evaluating the expression inside the radical: $3\left(\frac{3}{4}\right) = \frac{9}{4}$ .

3\left(\frac{3}{4}\right) = \frac{9}{4}

Now, we can substitute this simplified expression back into the equation:

$\sqrt[\frac{1}{3}]{\frac{9}{4}} = \frac{x}{4}$

Evaluating the Radical Expression

To evaluate the radical expression, we need to raise the number inside the radical to the power of the fractional exponent. In this case, the fractional exponent is $\frac{1}{3}$ , so we need to raise $\frac{9}{4}$ to the power of $\frac{1}{3}$ .

\left(\frac{9}{4}\right)^{\frac{1}{3}} = \sqrt[\frac{1]{3}]{\frac{9}{4}}

Using a calculator or a mathematical software, we can evaluate this expression:

$\left(\frac{9}{4}\right)^{\frac{1}{3}} \approx 1.1447$

Solving for $x$

Now that we have evaluated the radical expression, we can substitute this value back into the equation:

$1.1447 = \frac{x}{4}$

To solve for $x$ , we can multiply both sides of the equation by $4$ :

$x \approx 1.1447 \times 4$

$x \approx 4.5788$

Conclusion

In this article, we simplified and solved for $x$ in the equation $\sqrt[\frac{1}{3}]{3\left(\frac{1}{2}+\frac{1}{4}\right)} = \frac{x}{4}$ . We started by evaluating the expression inside the radical, then simplified the radical expression, and finally solved for $x$ . The final answer is $x \approx 4.5788$ .

Frequently Asked Questions

Q: What is the fractional exponent in the equation? A: The fractional exponent is $\frac{1}{3}$ .
Q: How do we evaluate the radical expression? A: We raise the number inside the radical to the power of the fractional exponent.
Q: How do we solve for $x$ ? A: We multiply both sides of the equation by $4$ .

Final Answer

The final answer is $\boxed{4.5788}$ .

Understanding the Problem

Evaluating the Expression Inside the Radical

\frac{1}{2}+\frac{1}{4} = \frac{2}{4}+\frac{1}{4} = \frac{3}{4}

Now, we can substitute this simplified expression back into the original equation:

$\sqrt[\frac{1}{3}]{3\left(\frac{3}{4}\right)} = \frac{x}{4}$

Simplifying the Radical Expression

Next, we need to simplify the radical expression. We can start by evaluating the expression inside the radical: $3\left(\frac{3}{4}\right) = \frac{9}{4}$ .

3\left(\frac{3}{4}\right) = \frac{9}{4}

Now, we can substitute this simplified expression back into the equation:

$\sqrt[\frac{1}{3}]{\frac{9}{4}} = \frac{x}{4}$

Evaluating the Radical Expression

\left(\frac{9}{4}\right)^{\frac{1}{3}} = \sqrt[\frac{1]{3}]{\frac{9}{4}}

Using a calculator or a mathematical software, we can evaluate this expression:

$\left(\frac{9}{4}\right)^{\frac{1}{3}} \approx 1.1447$

Solving for $x$

Now that we have evaluated the radical expression, we can substitute this value back into the equation:

$1.1447 = \frac{x}{4}$

To solve for $x$ , we can multiply both sides of the equation by $4$ :

$x \approx 1.1447 \times 4$

$x \approx 4.5788$

Q&A

Q: What is the fractional exponent in the equation?

A: The fractional exponent is $\frac{1}{3}$ .

Q: How do we evaluate the radical expression?

A: We raise the number inside the radical to the power of the fractional exponent.

Q: How do we solve for $x$ ?

A: We multiply both sides of the equation by $4$ .

Q: What is the common denominator for the fractions $\frac{1}{2}$ and $\frac{1}{4}$ ?

A: The common denominator is $4$ .

Q: How do we simplify the expression inside the radical?

A: We find a common denominator for the fractions and rewrite them as equivalent fractions.

Q: What is the value of $x$ ?

A: The value of $x$ is approximately $4.5788$ .

Q: How do we use a calculator or mathematical software to evaluate the radical expression?

A: We enter the expression into the calculator or software and evaluate it.

Q: What is the final answer?

A: The final answer is $\boxed{4.5788}$ .

Conclusion

Final Answer

The final answer is $\boxed{4.5788}$ .