Simplifying $216^{-2/3} imes 0.16^{-3/2}$: A Step-by-Step Guide

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Hey guys! Today, we're diving into the fascinating world of exponents and fractions to tackle a seemingly complex math problem. Don't worry, we'll break it down step-by-step so it's super easy to follow. Our mission? To simplify the expression $216^{-2/3} imes 0.16^{-3/2}$. Buckle up, because we're about to make math magic happen!

Understanding the Basics of Exponents

Before we jump into the main problem, let's quickly recap some exponent basics. Exponents are a shorthand way of showing repeated multiplication. For instance, $2^3$ means 2 multiplied by itself three times (2 * 2 * 2 = 8). But what happens when we encounter fractional or negative exponents? That's where things get interesting!

Fractional Exponents

A fractional exponent like $a^{m/n}$ represents both a power and a root. The numerator (m) indicates the power to which the base (a) is raised, and the denominator (n) indicates the root to be taken. In simpler terms, $a^{m/n}$ is the same as taking the nth root of a and then raising it to the power of m. Mathematically, we can express it as:

$a^{m/n} = (${ \sqrt[n]{a} }$ )^m$

Let's illustrate this with an example. Consider $8^{2/3}$. Here, the denominator is 3, so we need to find the cube root of 8, which is 2 (since 2 * 2 * 2 = 8). Then, we raise this result to the power of the numerator, which is 2. So, we have $2^2 = 4$. Therefore, $8^{2/3} = 4$. Understanding fractional exponents is crucial because it allows us to handle roots and powers in a single, elegant expression.

Negative Exponents

Now, let's talk about negative exponents. A negative exponent indicates the reciprocal of the base raised to the positive exponent. In other words, $a^{-n}$ is equal to $1/a^n$. This might seem a bit abstract, but it's a powerful rule that helps us deal with expressions where the exponent is negative. For example, $2^{-3}$ means $1/2^3$, which is $1/(2 * 2 * 2) = 1/8$. Essentially, a negative exponent tells us to move the base to the denominator (or vice versa) and change the sign of the exponent.

Combining these two concepts, we can tackle expressions with both fractional and negative exponents. For instance, if we have $a^{-m/n}$, we can rewrite it as $1/a^{m/n}$. Then, we apply the fractional exponent rule as before. This combination of rules is essential for simplifying complex expressions and is a cornerstone of many mathematical manipulations.

Breaking Down $216^{-2/3}$

Alright, let's get our hands dirty with the first part of the expression: $216^{-2/3}$. Remember our exponent rules? This looks like a perfect candidate for applying both the negative and fractional exponent rules. Let’s break it down step-by-step.

Step 1: Dealing with the Negative Exponent

The first thing we notice is the negative exponent, $-2/3$. As we discussed earlier, a negative exponent means we need to take the reciprocal. So, we can rewrite $216^{-2/3}$ as $1/216^{2/3}$. This step is crucial because it transforms the negative exponent into a positive one, making the subsequent calculations much easier. By taking the reciprocal, we shift the base (216) from the numerator to the denominator, and the exponent becomes positive.

Step 2: Tackling the Fractional Exponent

Now we have $1/216^{2/3}$. The fractional exponent $2/3$ tells us to take the cube root of 216 and then square the result. Why cube root first? Because finding the root first usually results in smaller numbers, which are easier to work with. Think of it as making the problem more manageable before we apply the power.

So, let’s find the cube root of 216. What number, when multiplied by itself three times, gives us 216? If you're familiar with your cubes, you'll know that 6 * 6 * 6 = 216. Therefore, the cube root of 216 is 6. We can write this as

${ \sqrt[3]{216} = 6 }$

Step 3: Squaring the Result

We've found the cube root; now it’s time to square it. We take the result from the previous step, which was 6, and square it. Squaring a number means multiplying it by itself, so we have:

${ 6^2 = 6 imes 6 = 36 }$

Step 4: Putting it All Together

Now, let's piece everything back together. We started with $216^{-2/3}$, rewrote it as $1/216^{2/3}$, found the cube root of 216 to be 6, and then squared 6 to get 36. So, $216^{2/3}$ is 36. But remember, we took the reciprocal at the beginning, so we have:

{ 216^{-2/3} = rac{1}{216^{2/3}} = rac{1}{36} }

And there you have it! We've successfully simplified $216^{-2/3}$ to $1/36$. This systematic approach, breaking down the problem into smaller, manageable steps, is key to conquering these types of expressions. Next up, we'll tackle the second part of the original expression.

Decoding $0.16^{-3/2}$

Now, let's tackle the second part of our original expression: $0.16^{-3/2}$. This might look a little intimidating because we're dealing with a decimal, but fear not! We can handle this just like we handled the previous term. The key here is to convert the decimal to a fraction, making it easier to work with. Trust me, it's a game-changer!

Step 1: Converting the Decimal to a Fraction

First things first, let's convert 0.16 into a fraction. We can express 0.16 as 16/100. This is because 0.16 represents sixteen hundredths. Now, we can simplify this fraction by finding the greatest common divisor (GCD) of 16 and 100. The GCD is 4, so we divide both the numerator and the denominator by 4:

{ rac{16}{100} = rac{16 ext{ ÷ } 4}{100 ext{ ÷ } 4} = rac{4}{25} }

So, 0.16 is equivalent to 4/25. This conversion is super helpful because fractions are much easier to manipulate when dealing with exponents and roots. Now we can rewrite our expression as $(4/25)^{-3/2}$.

Step 2: Dealing with the Negative Exponent (Again!)

Just like before, we have a negative exponent, -3/2. This means we need to take the reciprocal of the base. Remember, taking the reciprocal of a fraction means flipping the numerator and the denominator. So, we rewrite $(4/25)^{-3/2}$ as $(25/4)^{3/2}$. Notice how the negative sign in the exponent disappears once we take the reciprocal. This step is crucial for simplifying the expression further.

Step 3: Tackling the Fractional Exponent

Now we have $(25/4)^{3/2}$. The fractional exponent 3/2 tells us to take the square root (because the denominator is 2) and then cube the result (because the numerator is 3). Remember, it's often easier to find the root before raising to the power, as it keeps the numbers smaller and more manageable.

First, let's find the square root of 25/4. The square root of a fraction is the square root of the numerator divided by the square root of the denominator:

${ \[ \sqrt{\frac{25}{4}} = \frac{\sqrt{25}}{\sqrt{4}} = \frac{5}{2} }$

So, the square root of 25/4 is 5/2. This step simplifies our expression significantly, paving the way for the next operation.

Step 4: Cubing the Result

Next, we need to cube the result we obtained in the previous step. Cubing a fraction means raising both the numerator and the denominator to the power of 3:

${ \[ \left( \frac{5}{2} \right)^3 = \frac{5^3}{2^3} = \frac{5 imes 5 imes 5}{2 imes 2 imes 2} = \frac{125}{8} }$

So, $(5/2)^3$ is 125/8. This is the simplified form of $(25/4)^{3/2}$.

Step 5: Putting it All Together (Again!)

Let's recap our journey with $0.16^{-3/2}$. We converted 0.16 to 4/25, rewrote the expression as $(4/25)^{-3/2}$, took the reciprocal to get $(25/4)^{3/2}$, found the square root to be 5/2, and then cubed it to get 125/8. Therefore:

${ 0.16^{-3/2} = \frac{125}{8} }$

Fantastic! We've successfully simplified $0.16^{-3/2}$ to 125/8. Now that we've conquered both parts of the original expression, we're ready to combine our results and find the final answer.

Combining the Simplified Terms

We've done the heavy lifting! We simplified $216^{-2/3}$ to $1/36$ and $0.16^{-3/2}$ to $125/8$. Now, the final step is to multiply these simplified terms together to get our answer. This is where all our hard work pays off, and we see the elegance of simplifying complex expressions.

Step 1: Multiplying the Fractions

We need to multiply $1/36$ by $125/8$. Multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together:

${ \[ \frac{1}{36} imes \frac{125}{8} = \frac{1 imes 125}{36 imes 8} = \frac{125}{288} }$

So, when we multiply the two fractions, we get $125/288$. This might look like our final answer, but it's always a good idea to check if we can simplify the fraction further.

Step 2: Checking for Simplification

To simplify a fraction, we need to see if there are any common factors between the numerator (125) and the denominator (288). In other words, is there a number that divides both 125 and 288 without leaving a remainder?

The prime factors of 125 are 5 * 5 * 5, and the prime factors of 288 are 2 * 2 * 2 * 2 * 2 * 3 * 3. Looking at these prime factorizations, we can see that there are no common factors between 125 and 288. This means that the fraction $125/288$ is already in its simplest form. Sometimes, you might find a common factor and need to divide both the numerator and the denominator by that factor to simplify. But in this case, we're good to go!

Step 3: The Grand Finale

We've reached the end of our journey! We started with a seemingly complicated expression, broke it down into manageable parts, simplified each part using exponent rules and fraction manipulations, and finally, combined the results. So, the simplified form of $216^{-2/3} imes 0.16^{-3/2}$ is:

${ \[ \frac{125}{288} }$

Conclusion: You Nailed It!

There you have it, guys! We've successfully simplified the expression $216^{-2/3} imes 0.16^{-3/2}$ to $125/288$. How awesome is that? Remember, the key to tackling these kinds of problems is to break them down into smaller, more manageable steps. Understand the rules of exponents, especially fractional and negative exponents, and don't be afraid to convert decimals to fractions when it makes things easier.

By following this step-by-step approach, you can conquer even the most intimidating mathematical expressions. Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!