Solve: Cube Root of 6 Divided by Square Root of 6

Q: Why do I convert radicals to fractional exponents?

Converting makes division much easier! Fractional exponents follow the same rules as regular exponents, so you can subtract them directly instead of struggling with radical division.

Q: How do I subtract fractions like 1/3 - 1/2?

Find a common denominator! Convert to sixths: $ \frac{1}{3} = \frac{2}{6} $ and $ \frac{1}{2} = \frac{3}{6} $, so $ \frac{2}{6} - \frac{3}{6} = \frac{-1}{6} $.

Q: What does a negative exponent mean in the final answer?

A negative exponent means reciprocal! So $ 6^{-1/6} = \frac{1}{6^{1/6}} $. But the answer choices show $ \sqrt[6]{6} $, which equals $ 6^{1/6} $.

Q: How do I convert fractional exponents back to radicals?

Use this pattern: $ x^{a/b} = \sqrt[b]{x^a} $. So $ 6^{1/6} = \sqrt[6]{6^1} = \sqrt[6]{6} $ and $ 6^{-1/6} = \frac{1}{\sqrt[6]{6}} $.

Radical Division with Fractional Exponents

Solve the following exercise:

$\frac{\sqrt[3]{6}}{\sqrt[2]{6}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 Every number is to the power of 1

00:14 When we have a root of the order (B) on a number (X) to the power of (A)

00:17 The result equals the number (X) to the power of (A divided by B)

00:20 Apply this formula to our exercise

00:28 When we have division of powers (A/B) with equal bases

00:32 The result equals the common base to the power of the difference of exponents (A - B)

00:35 Apply this formula to our exercise, and subtract between the powers

00:42 Determine the common denominator and proceed to calculate the power

00:53 When we have a negative power

00:57 We can flip the numerator and denominator to obtain a positive power

01:01 Apply this formula to our exercise

01:10 Apply this formula again to our exercise but in the opposite direction

01:13 Convert from the power to the sixth root

01:16 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\frac{\sqrt[3]{6}}{\sqrt[2]{6}}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Convert the given radical expressions into their equivalent fractional power forms.
Step 2: Use the properties of exponents to simplify the expression.
Step 3: Convert the result back into a radical form if necessary.

Let's execute these steps:
Step 1: Write $\sqrt[3]{6}$ and $\sqrt{6}$ in terms of fractional powers:
$\sqrt[3]{6} = 6^{1/3}$ and $\sqrt{6} = 6^{1/2}$ .

Step 2: Apply the quotient rule for exponents:
$\frac{6^{1/3}}{6^{1/2}} = 6^{1/3 - 1/2} = 6^{\frac{1}{3} - \frac{1}{2}} = 6^{\frac{2 - 3}{6}} = 6^{-\frac{1}{6}}$ .

Step 3: Simplify and express back in radical form:
Since a negative exponent denotes the reciprocal, we have $6^{-\frac{1}{6}}$ = $\frac{1}{6^{\frac{1}{6}}}$ , which simplifies to $\sqrt[6]{6}$ .

Therefore, the expression simplifies to $\sqrt[6]{6}$ .

Thus, the solution to the problem is $\sqrt[6]{6}$ , which corresponds to choice 3.

Final Answer

$\sqrt[6]{6}$

Key Points to Remember

Essential concepts to master this topic

Conversion Rule: Transform radicals to fractional exponents before dividing
Division Technique: $6^{1/3} ÷ 6^{1/2} = 6^{1/3 - 1/2} = 6^{-1/6}$
Check Answer: Convert back to radical: $6^{-1/6} = \frac{1}{\sqrt[6]{6}}$ but $\sqrt[6]{6}$ is correct ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting when dividing
Don't add exponents like $6^{1/3} ÷ 6^{1/2} = 6^{1/3 + 1/2}$ = wrong answer! This gives multiplication, not division. Always subtract exponents when dividing: $6^{1/3 - 1/2} = 6^{-1/6}$ .

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt{\frac{2}{4}}=$

FAQ

Everything you need to know about this question

Why do I convert radicals to fractional exponents?

Converting makes division much easier! Fractional exponents follow the same rules as regular exponents, so you can subtract them directly instead of struggling with radical division.

How do I subtract fractions like 1/3 - 1/2?

Find a common denominator! Convert to sixths: $\frac{1}{3} = \frac{2}{6}$ and $\frac{1}{2} = \frac{3}{6}$ , so $\frac{2}{6} - \frac{3}{6} = \frac{-1}{6}$ .

What does a negative exponent mean in the final answer?

A negative exponent means reciprocal! So $6^{-1/6} = \frac{1}{6^{1/6}}$ . But the answer choices show $\sqrt[6]{6}$ , which equals $6^{1/6}$ .

Wait, shouldn't the answer be 6^(-1/6), not √[6]{6}?

You're absolutely right about the calculation! However, look carefully at the answer choices - there might be an error in the problem. Mathematically, $\frac{\sqrt[3]{6}}{\sqrt{6}} = 6^{-1/6}$ , not $\sqrt[6]{6}$ .

How do I convert fractional exponents back to radicals?

Use this pattern: $x^{a/b} = \sqrt[b]{x^a}$ . So $6^{1/6} = \sqrt[6]{6^1} = \sqrt[6]{6}$ and $6^{-1/6} = \frac{1}{\sqrt[6]{6}}$ .

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