Math Workbook: Powers and Roots - Practice Exercises with Answers

Let’s solve the expression step by step:

We are given:

\[
\frac{\sqrt{x} \cdot \sqrt[4]{x}}{\sqrt[3]{x}} = x^{\frac{3}{4} - \frac{1}{3}}
\]

Wait — actually, looking at the image again, it seems like the right-hand side is written as \( x^{\frac{3}{4} - \frac{1}{3}} \), but that might be a typo or misalignment. Let me check what the left-hand side simplifies to, and then see if it matches the exponent on the right.

Actually, let’s re-read the problem carefully from the image:

The equation shown is:

\[
\frac{\sqrt{x} \cdot \sqrt[4]{x}}{\sqrt[3]{x}} = x^{\frac{3}{4} - \frac{1}{3}}
\]

But that doesn’t look right — because the exponents on the left should add and subtract based on root rules, and we can compute them directly.

Let me convert all roots to fractional exponents:

- \( \sqrt{x} = x^{1/2} \)
- \( \sqrt[4]{x} = x^{1/4} \)
- \( \sqrt[3]{x} = x^{1/3} \)

So the left-hand side becomes:

\[
\frac{x^{1/2} \cdot x^{1/4}}{x^{1/3}}
\]

When multiplying powers with the same base, add the exponents:

Numerator: \( x^{1/2 + 1/4} = x^{(2/4 + 1/4)} = x^{3/4} \)

Now divide by \( x^{1/3} \): subtract the exponent:

\( x^{3/4 - 1/3} \)

Now compute \( \frac{3}{4} - \frac{1}{3} \):

Find common denominator: 12

\( \frac{3}{4} = \frac{9}{12} \), \( \frac{1}{3} = \frac{4}{12} \)

So: \( \frac{9}{12} - \frac{4}{12} = \frac{5}{12} \)

Therefore, the entire expression simplifies to:

\[
x^{5/12}
\]

But in the image, the right-hand side is written as \( x^{\frac{3}{4} - \frac{1}{3}} \), which is correct as an intermediate step — but not simplified. However, since the question appears to be asking us to verify or simplify, and the final simplified form is \( x^{5/12} \), I think that’s what we’re supposed to find.

Wait — maybe the image is showing that the left side equals \( x^{\frac{3}{4} - \frac{1}{3}} \), and perhaps the student is meant to simplify that exponent? Or maybe there's a mistake in how it's written?

Looking again: The image shows:

Left side: \( \frac{\sqrt{x} \cdot \sqrt[4]{x}}{\sqrt[3]{x}} \)

Right side: \( x^{\frac{3}{4} - \frac{1}{3}} \)

But as we just calculated, the left side simplifies to \( x^{3/4 - 1/3} = x^{5/12} \), so the equation is correct as written — it’s just that the exponent on the right is not simplified.

However, if this is a homework problem, likely the task is to simplify the entire expression — meaning, write it as a single power of x with a simplified exponent.

So let’s do that.

Step-by-step:

1. Rewrite all radicals as exponents:
- √x = x^(1/2)
- ⁺√x = x^(1/4) [fourth root]
- ∛x = x^(1/3)

2. Multiply numerator: x^(1/2) * x^(1/4) = x^(1/2 + 1/4) = x^(3/4)

3. Divide by denominator: x^(3/4) / x^(1/3) = x^(3/4 - 1/3)

4. Compute 3/4 - 1/3:
- LCD of 4 and 3 is 12
- 3/4 = 9/12
- 1/3 = 4/12
- 9/12 - 4/12 = 5/12

Final simplified form: x^(5/12)

So the answer is \( x^{5/12} \)

I double-checked the arithmetic: 3/4 minus 1/3 is indeed 5/12. Yes.

Also, note: the image may have intended to show the steps, and the right-hand side is part of the process — but the fully simplified answer is x to the 5/12.

Since the user said “solve the problem accurately”, and the problem shown is an equation that is technically true (because both sides equal x^{5/12}), but likely the expectation is to simplify the left-hand side to its simplest exponential form.

In many textbooks, when they show such expressions, they want you to combine into a single exponent.

So I will go with \( x^{5/12} \) as the final simplified answer.

Final Answer:
\[
x^{\frac{5}{12}}
\]