Equivalent Expression Of $\frac{\sqrt{4}}{\sqrt[3]{2}}$: Math Guide

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Hey everyone! Today, we are diving into a math problem that involves simplifying expressions with radicals. Specifically, we're going to figure out which expression is equivalent to $\frac{\sqrt{4}}{\sqrt[3]{2}}$. This type of problem is common in algebra and often appears in standardized tests, so understanding how to tackle it is super useful. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into the solution, let’s break down the expression we're dealing with: $\frac{\sqrt{4}}{\sqrt[3]{2}}$.

• Radicals and Roots: The expression involves square roots ($\sqrt{}$) and cube roots ($\sqrt[3]{}$). Remember, a square root of a number x ($\sqrt{x}$) is a value that, when multiplied by itself, gives you x. For example, $\sqrt{4}$ is 2 because 2 * 2 = 4. A cube root of a number y ($\sqrt[3]{y}$) is a value that, when multiplied by itself three times, gives you y. For example, $\sqrt[3]{8}$ is 2 because 2 * 2 * 2 = 8.
• Fractions: The expression is a fraction, which means we are dividing the numerator (the top part) by the denominator (the bottom part). In our case, we are dividing $\sqrt{4}$ by $\sqrt[3]{2}$.
• Equivalence: We need to find an expression that simplifies to the same value as $\frac{\sqrt{4}}{\sqrt[3]{2}}$. This might involve rewriting the radicals, simplifying the fraction, or using exponent rules.

To find the equivalent expression, we will need to manipulate the given expression using mathematical rules and properties. Let's dive into the step-by-step solution to make it crystal clear.

Step-by-Step Solution

Okay, let’s solve this step by step. Our goal is to simplify $\frac{\sqrt{4}}{\sqrt[3]{2}}$ and see which of the given options matches our simplified form.

Step 1: Simplify the Square Root

First, let's simplify the square root in the numerator. We know that $\sqrt{4} = 2$ because 2 * 2 = 4. So, we can rewrite the expression as:

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

Step 2: Rationalize the Denominator

The next step is to get rid of the cube root in the denominator. This process is called rationalizing the denominator. To do this, we need to multiply both the numerator and the denominator by a value that will turn the cube root in the denominator into a whole number.

Think about it: what do we need to multiply $\sqrt[3]{2}$ by to get a perfect cube? We need two more factors of 2, so we'll multiply by $\sqrt[3]{2^2}$ which is $\sqrt[3]{4}$. This will give us $\sqrt[3]{2} * \sqrt[3]{4} = \sqrt[3]{8} = 2$.

So, let's multiply both the numerator and the denominator by $\sqrt[3]{4}$:

$$\frac{2}{\sqrt[3]{2}} * \frac{\sqrt[3]{4}}{\sqrt[3]{4}} = \frac{2\sqrt[3]{4}}{\sqrt[3]{2*4}} = \frac{2\sqrt[3]{4}}{\sqrt[3]{8}}$$

Step 3: Simplify the Cube Root in the Denominator

Now, we simplify the cube root in the denominator. We know that $\sqrt[3]{8} = 2$ because 2 * 2 * 2 = 8. So, we have:

$$\frac{2\sqrt[3]{4}}{2}$$

Step 4: Simplify the Fraction

We can now simplify the fraction by canceling out the 2 in the numerator and the denominator:

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

Step 5: Rewrite the Cube Root

Now, let's rewrite $\sqrt[3]{4}$ in terms of exponents to match the format of the answer choices. We know that $4 = 2^2$, so we can write:

$$\sqrt[3]{4} = \sqrt[3]{2^2} = 2^{\frac{2}{3}}$$

Step 6: Convert to a Common Root

Looking at the answer choices, we see that they all have a 12th root. So, let's convert our expression to have a 12th root. To do this, we need to find an equivalent fraction for $\frac{2}{3}$ with a denominator of 12. We can multiply both the numerator and the denominator by 4:

$$\frac{2}{3} * \frac{4}{4} = \frac{8}{12}$$

So, we can rewrite our expression as:

$$2^{\frac{2}{3}} = 2^{\frac{8}{12}}$$

Now, we can convert this back to radical form:

$$2^{\frac{8}{12}} = \sqrt[12]{2^8}$$

Step 7: Calculate $2^8$

Let's calculate $2^8$:

$$2^8 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 256$$

So, our expression becomes:

$$\sqrt[12]{256}$$

Step 8: Match with the Answer Choices

Now, let’s look at the answer choices and see which one matches $\sqrt[12]{256}$.

• A. $\frac{\sqrt[12]{27}}{2}$
• B. $\frac{\sqrt[4]{24}}{2}$
• C. $\frac{\sqrt[12]{55296}}{2}$
• D. $\frac{\sqrt[12]{177147}}{3}$

None of these look quite right yet. We need to see if we can manipulate our expression further to match one of the options. Let's try to rewrite $\sqrt[12]{256}$ in a form that involves division.

Step 9: Further Simplification

We can rewrite 256 as $2^8$. We want to see if we can express this in a form that matches one of the denominators in the answer choices (2 or 3). Let's try dividing by $2^{12}$ inside the root and multiplying by 2 outside the root:

$$\sqrt[12]{256} = \frac{\sqrt[12]{256 * 2^{12}}}{2} = \frac{\sqrt[12]{2^8 * 2^{12}}}{2} = \frac{\sqrt[12]{2^{20}}}{2}$$

This doesn't seem to be getting us closer to any of the answer choices. Let's try a different approach. We know that $256 = 2^8$, so let's try to find a common factor that we can take out of the 12th root.

We can rewrite $\sqrt[12]{256}$ as:

$$\sqrt[12]{2^8} = 2^{\frac{8}{12}} = 2^{\frac{2}{3}}$$

Now, let's look at the answer choices again. We need to get this into a form that has a 12th root and a denominator.

Step 10: Matching with Answer Choice C

Let's try to manipulate our expression to match answer choice C: $\frac{\sqrt[12]{55296}}{2}$.

We want to see if we can rewrite $\sqrt[12]{256}$ in the form $\frac{\sqrt[12]{x}}{2}$. To do this, we can multiply both sides by 2:

$$2\sqrt[12]{256} = \sqrt[12]{x}$$

Now, raise both sides to the 12th power:

$$(2\sqrt[12]{256})^{12} = x$$

$$2^{12} * 256 = x$$

$$4096 * 256 = x$$

$$x = 1048576$$

This isn't matching up with 55296. Let's try another approach.

Let's go back to our simplified form $\sqrt[3]{4}$ and try to manipulate it to match answer choice C.

We want to see if $\sqrt[3]{4}$ is equal to $\frac{\sqrt[12]{55296}}{2}$. Let's rewrite $\sqrt[3]{4}$ as $2^{\frac{2}{3}}$ and the answer choice C as follows:

$$\frac{\sqrt[12]{55296}}{2} = \frac{(55296)^{\frac{1}{12}}}{2}$$

We want to check if:

$$2^{\frac{2}{3}} = \frac{(55296)^{\frac{1}{12}}}{2}$$

Multiply both sides by 2:

$$2 * 2^{\frac{2}{3}} = (55296)^{\frac{1}{12}}$$

$$2^{\frac{5}{3}} = (55296)^{\frac{1}{12}}$$

Raise both sides to the power of 12:

$$(2^{\frac{5}{3}})^{12} = 55296$$

$$2^{20} = 55296$$

$$1048576 = 55296$$

This is not true, so answer choice C is not correct.

Step 11: Correcting an Error and Finding the Right Path

Okay, guys, it seems like we made a mistake somewhere in our calculations. Let’s backtrack and re-evaluate our steps. We got to $\sqrt[3]{4}$, which is correct. Now, let's try converting the answer choices to simpler forms to compare them accurately.

Let's re-examine answer choice C: $\frac{\sqrt[12]{55296}}{2}$. We need to determine if $\sqrt[3]{4}$ is equivalent to this.

We can rewrite $\sqrt[3]{4}$ as $2^{\frac{2}{3}}$. Now let’s work on answer choice C. We want to see if:

$$\sqrt[3]{4} = \frac{\sqrt[12]{55296}}{2}$$

Multiply both sides by 2:

$$2 * \sqrt[3]{4} = \sqrt[12]{55296}$$

Raise both sides to the power of 12:

$$(2 * \sqrt[3]{4})^{12} = 55296$$

$$2^{12} * (4^{\frac{1}{3}})^{12} = 55296$$

$$2^{12} * 4^4 = 55296$$

$$4096 * 256 = 55296$$

Here’s where we made a mistake before! 4096 * 256 is not 55296. It’s 1048576. Let’s correct this and re-evaluate 55296.

If we divide 55296 by 4096 ($2^{12}$), we get:

$$\frac{55296}{4096} = 13.5$$

This doesn’t give us a clean integer, so let's rethink our approach again.

Let’s go back to the basics and prime factorize 55296:

55296 = 2^{12} * rac{27}{2^3}

Oh! We see another error. Correct prime factorization of $55296 = 2^{12} * 3^{3}$.

So, $\sqrt[12]{55296} = \sqrt[12]{2^{12} * 3^3} = 2 * \sqrt[12]{3^3} = 2 * \sqrt[4]{3}$

Therefore, $\frac{\sqrt[12]{55296}}{2} = \sqrt[4]{3}$

Now, is $\sqrt[3]{4} = \sqrt[4]{3}$? Let's raise both sides to the power of 12 to see:

$(\sqrt[3]{4})^{12} = 4^4 = 256$ $(\sqrt[4]{3})^{12} = 3^3 = 27$

These are not equal, so answer choice C is still incorrect.

Step 12: Trying Answer Choice A

Let's analyze answer choice A: $\frac{\sqrt[12]{27}}{2}$. We want to determine if $\sqrt[3]{4}$ is equivalent to this.

Rewrite $\sqrt[12]{27}$ as $\sqrt[12]{3^3} = 3^{\frac{3}{12}} = 3^{\frac{1}{4}} = \sqrt[4]{3}$. So, answer choice A is:

$$\frac{\sqrt[4]{3}}{2}$$

Now we want to check if:

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

Multiply both sides by 2:

$$2\sqrt[3]{4} = \sqrt[4]{3}$$

Raise both sides to the power of 12:

$$(2\sqrt[3]{4})^{12} = (\sqrt[4]{3})^{12}$$

$$2^{12} * 4^4 = 3^3$$

$$4096 * 256 = 27$$

This is clearly not correct, so answer choice A is incorrect.

Step 13: Analyzing Answer Choice D

Let's consider option D: $\frac{\sqrt[12]{177147}}{3}$.

Prime factorize 177147: $177147 = 3^{11}$. So, $\sqrt[12]{177147} = \sqrt[12]{3^{11}} = 3^{\frac{11}{12}}$

Then, option D is $\frac{3^{\frac{11}{12}}}{3} = 3^{\frac{11}{12} - 1} = 3^{\frac{-1}{12}}$ which cannot be equivalent to $\sqrt[3]{4}$. Hence, option D is incorrect.

Step 14: Going Back to Answer Choice B

We've ruled out A, C, and D. Let's re-examine option B: $\frac{\sqrt[4]{24}}{2}$.

We want to see if $\sqrt[3]{4} = \frac{\sqrt[4]{24}}{2}$. Multiply both sides by 2:

$$2\sqrt[3]{4} = \sqrt[4]{24}$$

Raise both sides to the power of 12:

$$(2\sqrt[3]{4})^{12} = (\sqrt[4]{24})^{12}$$

$$2^{12} * 4^4 = 24^3$$

$$4096 * 256 = 24^3$$

$$1048576 = 13824$$

Oops! This is incorrect as well. It seems we still missed something.

Step 15: A Crucial Insight and Final Solution

Alright, guys, we’ve been through a rollercoaster of calculations, and it's time to revisit our fundamental approach. We correctly simplified the original expression to $\sqrt[3]{4}$. The issue lies in converting this to match the answer choices directly. Let’s try a different tack.

Since $\sqrt[3]{4} = 2^{\frac{2}{3}}$, let's express the answer choices in exponential form and see if we can find a match.

• A. $\frac{\sqrt[12]{27}}{2}$: $\frac{(3^3)^{\frac{1}{12}}}{2} = \frac{3^{\frac{1}{4}}}{2}$
• B. $\frac{\sqrt[4]{24}}{2}$: $\frac{(2^3 * 3)^{\frac{1}{4}}}{2} = \frac{2^{\frac{3}{4}} * 3^{\frac{1}{4}}}{2} = 2^{-\frac{1}{4}} * 3^{\frac{1}{4}}$
• C. $\frac{\sqrt[12]{55296}}{2}$: We found $55296 = 2^{12} * 3^3$, so $\frac{(2^{12} * 3^3)^{\frac{1}{12}}}{2} = \frac{2 * 3^{\frac{1}{4}}}{2} = 3^{\frac{1}{4}}$
• D. $\frac{\sqrt[12]{177147}}{3}$: $\frac{(3^{11})^{\frac{1}{12}}}{3} = 3^{\frac{11}{12} - 1} = 3^{-\frac{1}{12}}$

None of these directly match $2^{\frac{2}{3}}$. However, let’s go back to choice B and meticulously check our conversion:

$$\frac{\sqrt[4]{24}}{2} = \frac{(2^3 * 3)^{\frac{1}{4}}}{2} = \frac{2^{\frac{3}{4}} * 3^{\frac{1}{4}}}{2^1} = 2^{\frac{3}{4} - 1} * 3^{\frac{1}{4}} = 2^{-\frac{1}{4}} * 3^{\frac{1}{4}}$$

This is still not matching. Let’s try another approach. Multiply numerator and denominator by $\sqrt[4]{2}$:

$$\frac{\sqrt[4]{24}}{2} = \frac{\sqrt[4]{24} * \sqrt[4]{2}}{2 * \sqrt[4]{2}} = \frac{\sqrt[4]{48}}{2 * \sqrt[4]{2}}$$

Still not seeming correct. It seems our simplification journey is not as smooth as we thought.

Guys, I'm incredibly sorry, but after meticulously reviewing all steps and calculations multiple times, there seems to be an issue with the provided answer choices. None of them appear to be mathematically equivalent to the simplified form of the given expression, which we correctly derived as $\sqrt[3]{4}$ or $2^{\frac{2}{3}}$.

It's possible that there's a typo in one of the options or that the question is designed to highlight the importance of careful simplification and recognizing non-equivalent expressions.

Key Takeaways

• Simplify Step by Step: Break down the problem into smaller, manageable steps. Simplify radicals, rationalize denominators, and use exponent rules.
• Convert to Common Roots: When comparing expressions with different roots, convert them to a common root to make comparison easier.
• Check Your Work: Always double-check your calculations to avoid errors.
• Don't Be Afraid to Backtrack: If you get stuck, go back and review your steps. There might be a small error that’s throwing everything off.
• Prime Factorization is Key: Always consider prime factorization to make your life easier.

Conclusion

Whew! We went on quite the math adventure today! Even though we didn't find a matching answer choice due to a potential issue with the options, we learned a ton about simplifying radical expressions. Remember, the process is just as important as the answer. Keep practicing, and you'll become a pro at these types of problems in no time! Keep up the great work, guys!