Solving 32^(2/5) - (1/100)^(-3/2) + 27^(4/3) A Step-by-Step Guide

Hey guys! Today, we're diving into a fun math problem that involves exponents and fractions. Don't worry, it's not as intimidating as it looks! We're going to break down the expression 32^(2/5) - (1/100)^(-3/2) + 27^(4/3) step by step, so you can understand exactly how to solve it. So, let's get started and conquer these exponents!

Breaking Down the Problem

When we first look at this problem, 32^(2/5) - (1/100)^(-3/2) + 27^(4/3), it might seem a bit overwhelming. But, the trick is to tackle it piece by piece. We have three main terms here, each involving exponents. Remember, exponents tell us how many times to multiply a number by itself, but when we have fractional exponents, it's like we're dealing with roots and powers combined. So, we're going to simplify each term individually, and then we'll put it all together. Think of it like building a puzzle – we'll assemble the individual pieces, and then we'll see the whole picture. Let's dive into the first term and see what we can discover!

Solving the First Term: 32^(2/5)

Alright, let's focus on our first term: 32^(2/5). This might look a bit tricky, but we can break it down using the rules of exponents. Remember that a fractional exponent like 2/5 means we're dealing with both a power and a root. The denominator (5) tells us which root to take, and the numerator (2) tells us which power to raise the result to. So, 32^(2/5) can be interpreted as the fifth root of 32, all raised to the power of 2. The fifth root of 32 is the number that, when multiplied by itself five times, equals 32. Think about it: 2 * 2 * 2 * 2 * 2 = 32. So, the fifth root of 32 is 2. Now, we need to raise this result (2) to the power of 2. That means 2 squared, which is 2 * 2 = 4. So, we've successfully simplified the first term: 32^(2/5) = 4. See? Not so scary when we take it one step at a time! Let's move on to the second term and see what challenges await us there.

Tackling the Second Term: (1/100)^(-3/2)

Now, let's move on to the second term: (1/100)^(-3/2). This one looks a little more complicated because we have a fraction (1/100) and a negative exponent (-3/2). But don't worry, we can handle it! First, let's deal with the negative exponent. Remember that a negative exponent means we need to take the reciprocal of the base. So, (1/100)^(-3/2) is the same as (100/1)^(3/2), which is simply 100^(3/2). Now we have a fractional exponent to deal with. Just like before, the denominator (2) tells us which root to take, and the numerator (3) tells us which power to raise the result to. So, 100^(3/2) means we need to find the square root of 100 and then raise it to the power of 3. The square root of 100 is 10 (since 10 * 10 = 100). Now, we raise 10 to the power of 3, which means 10 * 10 * 10 = 1000. So, we've simplified the second term: (1/100)^(-3/2) = 1000. We're making great progress! Let's move on to the final term and see how we can simplify it.

Simplifying the Third Term: 27^(4/3)

Okay, let's tackle the final term: 27^(4/3). Just like the previous terms, we have a fractional exponent here. The denominator (3) tells us we need to find the cube root of 27, and the numerator (4) tells us to raise the result to the power of 4. So, what's the cube root of 27? That's the number that, when multiplied by itself three times, equals 27. Think about it: 3 * 3 * 3 = 27. So, the cube root of 27 is 3. Now, we need to raise 3 to the power of 4. That means 3 * 3 * 3 * 3. Let's calculate that: 3 * 3 = 9, 9 * 3 = 27, and 27 * 3 = 81. So, 27^(4/3) = 81. We've successfully simplified all three terms! Now, the fun part: let's put it all together.

Putting It All Together: The Final Calculation

We've done the hard work of simplifying each term individually. Now, it's time to bring it all together and get our final answer. Remember our original expression: 32^(2/5) - (1/100)^(-3/2) + 27^(4/3). We found that 32^(2/5) = 4, (1/100)^(-3/2) = 1000, and 27^(4/3) = 81. So, we can substitute these values back into the original expression: 4 - 1000 + 81. Now, it's just a simple arithmetic problem. First, let's do 4 - 1000, which gives us -996. Then, we add 81 to that: -996 + 81 = -915. So, the final answer is -915. We did it! We successfully solved the expression by breaking it down into smaller, manageable steps. Remember, guys, when you encounter a complex problem, don't be afraid to break it down and tackle it piece by piece. You've got this!

Conclusion

So, there you have it! We've walked through the process of solving the expression 32^(2/5) - (1/100)^(-3/2) + 27^(4/3) step by step. We started by understanding the individual terms and how fractional exponents work. We then simplified each term separately, dealing with roots, powers, and negative exponents. Finally, we put it all together and performed the arithmetic to arrive at our final answer: -915. Remember, the key to tackling complex math problems is to break them down into smaller, more manageable parts. By understanding the rules of exponents and applying them methodically, you can solve even the trickiest problems. Keep practicing, and you'll become a math whiz in no time! And remember, math can actually be fun when you approach it with a positive attitude and a willingness to learn. Keep exploring, keep questioning, and keep solving!