Evaluate $25^{-3/2}$ As A Reduced Fraction

Hey math whizzes! Today, we're diving into a cool problem that'll test your knowledge of exponents and fractions. We're going to evaluate the expression $25^{-3/2}$ and make sure our final answer is a neat, reduced fraction. So grab your pencils, dust off those math brains, and let's get this done together!

Understanding the Exponent Rules, Guys!

Before we even touch the number 25, let's break down what that exponent $-3/2$ means. You know how exponents can be a bit tricky, right? Well, this one has a negative sign and a fraction, which means a couple of things are happening. First off, that negative sign in the exponent tells us we're dealing with a reciprocal. So, $a^{-n}$ is the same as $1/a^n$. This is a fundamental rule, and it's going to be super important for solving our problem. Don't forget it!

Secondly, that fractional exponent like $1/2$ or $3/2$ indicates roots. Specifically, an exponent of $1/2$ means taking the square root. So, if we have $a^{1/2}$, it's the same as $\sqrt{a}$. And if we have something like $a^{m/n}$, it means we're taking the nth root of 'a' and then raising that result to the mth power, or we can raise 'a' to the mth power and then take the nth root. It's usually easier to take the root first, then raise it to the power. So, $a^{m/n} = (\sqrt[n]{a})^m$. Keep these rules in mind, as they are the keys to unlocking this problem.

Tackling the Base: The Number 25

Now, let's look at our base number, which is 25. To make things easier, it's always a good idea to see if our base can be expressed as a power of a smaller number. In this case, 25 is a perfect square! We know that $25 = 5^2$. Seeing this connection will simplify our calculation immensely. Why? Because when we substitute $5^2$ for 25 in our original expression, we'll be able to use another awesome exponent rule: the power of a power rule. Remember, $(a^m)^n = a^{m \times n}$? This rule is our best friend when dealing with nested exponents like we have here.

So, instead of dealing with 25, we'll be working with $5^2$. This is a crucial step in simplifying the problem and making it much more manageable. It's all about breaking down the problem into smaller, more understandable parts. By recognizing that 25 is $5^2$, we've set ourselves up for a much smoother calculation. Don't underestimate the power of simplifying the base first, guys!

Putting It All Together: Step-by-Step Evaluation

Alright, let's combine everything we've learned and evaluate $25^{-3/2}$.

Step 1: Substitute the base. We know that $25 = 5^2$, so we can rewrite our expression as: $(5^2)^{-3/2}$.

Step 2: Apply the power of a power rule. Using the rule $(a^m)^n = a^{m \times n}$, we multiply the exponents: $5^{2 \times (-3/2)}$.

$2 \times (-3/2) = -3$. So, our expression simplifies to $5^{-3}$.

Step 3: Deal with the negative exponent. Now we have $5^{-3}$. Remember our rule for negative exponents? $a^{-n} = 1/a^n$. So, $5^{-3}$ becomes $1/5^3$.

Step 4: Calculate the remaining power. We need to calculate $5^3$. That's $5 \times 5 \times 5$. Well, $5 \times 5 = 25$, and $25 \times 5 = 125$.

So, $1/5^3 = 1/125$.

Step 5: Express as a reduced fraction. Our answer is $1/125$. Is this a reduced fraction? Yes! The only common factor between 1 and 125 is 1. So, we can't simplify it any further. Awesome!

Alternative Approach: Root First, Then Power

Just to show you guys another way to think about it, let's consider the fractional exponent $-3/2$ as a root and a power. Remember $a^{m/n} = (\sqrt[n]{a})^m$? In our case, $m = -3$ and $n = 2$. So, $25^{-3/2}$ can be thought of as $(\sqrt[2]{25})^{-3}$ or $(\sqrt{25})^{-3}$.

Step 1: Calculate the square root. The square root of 25, $\sqrt{25}$, is 5.

Step 2: Raise to the negative power. Now we have $5^{-3}$. This is exactly where we were in the previous method after simplifying the base. So, $5^{-3} = 1/5^3$.

Step 3: Calculate the final value. $1/5^3 = 1/125$.

See? We get the same answer! It really just depends on which exponent rule you find easiest to apply first. Both methods are totally valid, and understanding them helps solidify your grasp on exponent manipulation.

Why This Matters: Real-World Connections (Kind Of!)

Okay, so maybe you won't be calculating $25^{-3/2}$ every day for your grocery shopping. But understanding these exponent rules is super important in lots of areas of math and science. Think about compound interest, population growth, radioactive decay, or even how computer algorithms work. Many of these involve exponential functions, and being comfortable with fractional and negative exponents is key to understanding and manipulating those formulas.

When you're dealing with things like scientific notation, complex calculations in physics, or even just understanding growth rates in economics, these foundational exponent rules are the building blocks. Mastering them now will make tackling more advanced math concepts down the line a whole lot smoother. It’s like learning your ABCs before you can read a novel – essential stuff!

So, next time you see a problem like $25^{-3/2}$, don't get intimidated. Break it down, remember your exponent rules, and tackle it step by step. You guys are totally capable of mastering this!

Final Answer Check

We evaluated $25^{-3/2}$.

1. We rewrote 25 as $5^2$.
2. We used the power of a power rule to get $5^{2 \times (-3/2)} = 5^{-3}$.
3. We used the negative exponent rule to get $1/5^3$.
4. We calculated $5^3 = 125$.

Our final answer is 1/125, which is a reduced fraction. Mission accomplished!

Keep practicing these kinds of problems, and you'll become an exponent pro in no time. Happy calculating, everyone!