Equivalent Expressions: (5^(1/8) * 5^(3/8))^3

Hey math enthusiasts! Let's dive into a cool problem involving exponents and radicals. We're going to break down the expression $\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3$ and figure out which of the given options are equivalent. This is a fantastic way to sharpen our skills in manipulating exponents and understanding how they relate to radicals. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let’s make sure we understand what the question is asking. We have the expression $\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3$, and we need to determine which of the following options are equal to it:

• A. $5^{\frac{3}{2}}$
• B. $5^{\frac{9}{8}}$
• C. $\sqrt{5^3}$
• D. $(\sqrt[2]{5})^0$

To tackle this, we'll need to use the rules of exponents and simplify the original expression. Remember, when we multiply numbers with the same base, we add their exponents. And when we raise a power to a power, we multiply the exponents. These rules are our best friends in this scenario.

Step-by-Step Solution

Okay, let’s break it down step by step. First, we'll focus on simplifying the expression inside the parentheses:

$\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3$

Step 1: Combine the exponents inside the parentheses.

Since we're multiplying numbers with the same base (which is 5), we add the exponents:

$5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}} = 5^{\frac{1}{8} + \frac{3}{8}} = 5^{\frac{4}{8}}$

We can simplify the fraction $\frac{4}{8}$ to $\frac{1}{2}$, so we have:

$5^{\frac{1}{2}}$

Step 2: Apply the outer exponent.

Now, we raise this result to the power of 3:

$\left(5^{\frac{1}{2}}\right)^3$

When we raise a power to a power, we multiply the exponents:

$5^{\frac{1}{2} \cdot 3} = 5^{\frac{3}{2}}$

So, our simplified expression is $5^{\frac{3}{2}}$. Now, let's see which of the options match this.

Evaluating the Options

Let's go through each option and see if they are equivalent to $5^{\frac{3}{2}}$. This is where we get to put our thinking caps on and see how well we can manipulate exponents and radicals!

Option A: $5^{\frac{3}{2}}$

This one is straightforward! It's exactly what we got after simplifying our original expression. So, option A is definitely equivalent.

Option B: $5^{\frac{9}{8}}$

This looks similar, but the exponent is different. $\frac{9}{8}$ is not equal to $\frac{3}{2}$, so option B is not equivalent. To be sure, let's convert $\frac{3}{2}$ to have a denominator of 8: $\frac{3}{2} = \frac{12}{8}$. Clearly, $\frac{9}{8}$ and $\frac{12}{8}$ are different.

Option C: $\sqrt{5^3}$

This one involves a radical, so we need to remember how radicals and exponents are related. The square root of a number can be written as that number raised to the power of $\frac{1}{2}$. So, $\sqrt{5^3}$ can be rewritten as $(5^3)^{\frac{1}{2}}$. When we raise a power to a power, we multiply the exponents:

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

Aha! This is also equivalent to our simplified expression. So, option C is a match!

Option D: $(\sqrt[2]{5})^0$

Anything (except 0) raised to the power of 0 is 1. So, $(\sqrt[2]{5})^0 = 1$. This is definitely not equal to $5^{\frac{3}{2}}$, which is a value greater than 1. Therefore, option D is not equivalent.

Final Answer

Alright, we've done the math and evaluated all the options. The expressions equivalent to $\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3$ are:

• A. $5^{\frac{3}{2}}$
• C. $\sqrt{5^3}$

So, the correct answers are A and C. Give yourselves a pat on the back if you got that right!

Key Concepts Revisited

Let's quickly recap the key concepts we used to solve this problem. These are super important for mastering exponents and radicals, guys!

1. Product of Powers: When multiplying numbers with the same base, add the exponents. This is what we used in the first step: $a^m \cdot a^n = a^{m+n}$.
2. Power of a Power: When raising a power to a power, multiply the exponents. We used this rule when simplifying $(5^{\frac{1}{2}})^3$ and $(5^3)^{\frac{1}{2}}$: $(a^m)^n = a^{m \cdot n}$.
3. Fractional Exponents and Radicals: A fractional exponent represents a radical. Specifically, $a^{\frac{1}{n}}$ is the nth root of a, and $a^{\frac{m}{n}}$ is the nth root of $a^m$. We used this to convert between $\sqrt{5^3}$ and $5^{\frac{3}{2}}$.
4. Zero Exponent: Any non-zero number raised to the power of 0 is 1. This helped us quickly eliminate option D.

Understanding these rules is crucial for simplifying expressions and solving problems involving exponents and radicals. Keep practicing, and you'll become a pro in no time!

Practice Problems

Want to test your skills further? Here are a couple of practice problems similar to the one we just solved. Give them a try, and feel free to share your answers in the comments below!

1. Which expression(s) are equivalent to $\left(4^{\frac{2}{5}} \cdot 4^{\frac{3}{5}}\right)^2$?
2. Simplify the expression $\frac{(3^2)^{\frac{1}{2}} \cdot 3^{\frac{5}{2}}}{3^3}$.

Working through problems like these will solidify your understanding and make you more confident in tackling exponent and radical challenges. Math can be super fun when you get the hang of it!

Conclusion

So, there you have it! We've successfully navigated through this exponent problem, simplified the expression, and identified the equivalent options. Remember, the key to mastering these types of problems is understanding the rules of exponents and radicals and practicing regularly. Keep up the great work, guys, and I'll see you in the next math adventure!

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