Solve: Square Root of 4 Divided by Cube Root of 4

Exponent Rules with Fractional Powers

Solve the following exercise:

4243= \frac{\sqrt[2]{4}}{\sqrt[3]{4}}=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:06 Let's simplify this problem together!
00:09 Remember, each number is raised to the power of one.
00:16 When you have a root of order B, on X to the power of A,
00:25 this equals X to the power of A divided by B.
00:30 Let's apply this rule to our exercise.
00:37 If dividing powers A by B with the same base,
00:44 you get that base to the power of A minus B.
00:48 Now, let's use this formula and subtract the powers.
00:55 Find the common denominator, and calculate the new power.
01:02 And that's how we solve the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Solve the following exercise:

4243= \frac{\sqrt[2]{4}}{\sqrt[3]{4}}=

2

Step-by-step solution

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

  • Step 1: Convert the roots to exponent notation
  • Step 2: Apply the quotient of powers rule
  • Step 3: Simplify the expression

Now, let's work through each step:
Step 1: Convert the roots to exponent notation:
4=41/2\sqrt{4} = 4^{1/2} and 43=41/3\sqrt[3]{4} = 4^{1/3}.
Step 2: Calculate the quotient using the rule aman=am−n\frac{a^m}{a^n} = a^{m-n}:

41/241/3=4(1/2)−(1/3) \frac{4^{1/2}}{4^{1/3}} = 4^{(1/2) - (1/3)}

Step 3: Simplify the exponent:

1/2−1/3=36−26=16 1/2 - 1/3 = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}

Therefore, the expression simplifies to:

416 4^{\frac{1}{6}}

The correct answer is choice 1: 4164^{\frac{1}{6}}.

3

Final Answer

416 4^{\frac{1}{6}}

Key Points to Remember

Essential concepts to master this topic
  • Convert Roots: Change radicals to fractional exponents before dividing
  • Quotient Rule: aman=am−n \frac{a^m}{a^n} = a^{m-n} so 41/241/3=41/2−1/3 \frac{4^{1/2}}{4^{1/3}} = 4^{1/2 - 1/3}
  • Check Exponent: Verify 12−13=3−26=16 \frac{1}{2} - \frac{1}{3} = \frac{3-2}{6} = \frac{1}{6} ✓

Common Mistakes

Avoid these frequent errors
  • Dividing the radical numbers directly
    Don't divide 4÷43 \sqrt{4} ÷ \sqrt[3]{4} as 2 ÷ something = wrong approach! This ignores the different root types and leads to incorrect results. Always convert to exponential form first, then use exponent rules.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that is equal to the following:

\( \sqrt{a}:\sqrt{b} \)

FAQ

Everything you need to know about this question

Why can't I just calculate 4=2 \sqrt{4} = 2 and divide by the cube root?

+

While 4=2 \sqrt{4} = 2 , the cube root of 4 isn't a nice whole number! Converting to exponential form lets you use algebra rules that work with any base.

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

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Find a common denominator: 12−13=36−26=16 \frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} . The LCD of 2 and 3 is 6.

What does 41/6 4^{1/6} actually mean?

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It means the 6th root of 4, or 46 \sqrt[6]{4} . Any fractional exponent am/n a^{m/n} equals amn \sqrt[n]{a^m} .

Can I leave my answer as 41/6 4^{1/6} ?

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Yes! Exponential form is often the simplest way to express the answer. Converting to decimal form would give you an approximation, not the exact value.

What if the bases were different numbers?

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If you have different bases like 843 \frac{\sqrt{8}}{\sqrt[3]{4}} , you'd need to factor them into the same base first, or use logarithms for more complex problems.

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