Cube Root Of 64/27: A Step-by-Step Solution

Oct 13, 2025 by ADMIN 44 views

Hey guys! Let's dive into a fun little math problem today: finding the cube root of 64/27. It might sound intimidating, but trust me, it's easier than you think. We'll break it down step by step, so you'll be a pro in no time!

Understanding Cube Roots

Before we jump into the problem, let's quickly recap what a cube root actually is. When we talk about the cube root of a number, we're asking: "What number, when multiplied by itself three times, gives us the original number?" For example, the cube root of 8 is 2 because 2 * 2 * 2 = 8. Makes sense, right? Cube roots are the inverse operation of cubing a number. So, if $2^3 = 8$ , then $\sqrt[3]{8} = 2$ . This fundamental understanding is crucial as we approach our main problem.

Now, how do we approach finding the cube root of a fraction like 64/27? The trick here is to remember that we can take the cube root of the numerator and the denominator separately. This simplifies the problem and makes it much more manageable. Think of it as distributing the cube root operation across the fraction. So, $\sqrt[3]{\frac{64}{27}}$ becomes $\frac{\sqrt[3]{64}}{\sqrt[3]{27}}$ .

Why does this work? Well, consider a general fraction a/b. The cube root of a/b can be written as $(a/b)^{1/3}$ . Using exponent rules, this is equivalent to $a^{1/3} / b^{1/3}$ , which is the same as $\frac{\sqrt[3]{a}}{\sqrt[3]{b}}$ . This property is incredibly useful for simplifying expressions involving roots of fractions. Keep this in your back pocket—it’ll come in handy!

Breaking Down the Problem: $\sqrt[3]{\frac{64}{27}}$

Okay, now that we've got the basics down, let's tackle our specific problem: $\sqrt[3]{\frac{64}{27}}$ . Remember, we can split this up into finding the cube root of 64 and the cube root of 27 separately. So, we need to find $\sqrt[3]{64}$ and $\sqrt[3]{27}$ .

Finding the Cube Root of 64

First, let's find $\sqrt[3]{64}$ . We need to think: "What number, when multiplied by itself three times, equals 64?" Let's try a few numbers:

1 * 1 * 1 = 1 (too small)
2 * 2 * 2 = 8 (still too small)
3 * 3 * 3 = 27 (getting closer)
4 * 4 * 4 = 64 (bingo!)

So, the cube root of 64 is 4. We can write this as $\sqrt[3]{64} = 4$ . Keep in mind that knowing your perfect cubes (like 1, 8, 27, 64, 125, etc.) can speed up this process significantly. Recognizing these numbers can make these calculations much faster.

Finding the Cube Root of 27

Next, let's find $\sqrt[3]{27}$ . We're looking for a number that, when multiplied by itself three times, equals 27. We actually already tried this one out when we were finding the cube root of 64! Remember, 3 * 3 * 3 = 27. So, the cube root of 27 is 3. We can write this as $\sqrt[3]{27} = 3$ .

Putting It All Together

Now that we know $\sqrt[3]{64} = 4$ and $\sqrt[3]{27} = 3$ , we can put it all back together. Remember, we split up $\sqrt[3]{\frac{64}{27}}$ into $\frac{\sqrt[3]{64}}{\sqrt[3]{27}}$ . Now we can substitute the values we found:

$\frac{\sqrt[3]{64}}{\sqrt[3]{27}} = \frac{4}{3}$

So, the cube root of 64/27 is 4/3. That's it! We've solved the problem. Wasn't that fun?

Converting to a Mixed Number (Optional)

Sometimes, you might want to express your answer as a mixed number. A mixed number has a whole number part and a fractional part. To convert 4/3 to a mixed number, we divide 4 by 3.

4 divided by 3 is 1 with a remainder of 1.

So, 4/3 is equal to 1 and 1/3. We can write this as $1\frac{1}{3}$ . Both 4/3 and $1\frac{1}{3}$ are correct answers, but sometimes one form is preferred over the other, depending on the context.

Practice Problems

Want to test your skills? Try these practice problems:

$\sqrt[3]{\frac{8}{125}}$
$\sqrt[3]{\frac{216}{1000}}$
$\sqrt[3]{\frac{1}{64}}$

See if you can solve them using the same method we used above. Don't be afraid to take your time and break down each problem into smaller steps. Remember, practice makes perfect!

Tips and Tricks for Cube Roots

Here are a few tips and tricks to help you with cube roots:

Memorize Perfect Cubes: Knowing the first few perfect cubes (1, 8, 27, 64, 125, 216, 343, 512, 729, 1000) can save you a lot of time.
Prime Factorization: If you're struggling to find the cube root of a number, try prime factorization. Break the number down into its prime factors and look for groups of three identical factors.
Use a Calculator: If you're allowed to use a calculator, take advantage of it! Most calculators have a cube root function.
Estimation: If you don't have a calculator, try estimating. Think about which perfect cubes the number falls between.
Practice Regularly: The more you practice, the better you'll become at finding cube roots.

Real-World Applications of Cube Roots

You might be wondering, "When will I ever use cube roots in real life?" Well, cube roots actually have many practical applications in various fields.

Geometry: Cube roots are used to calculate the side length of a cube given its volume. For example, if you know the volume of a cube is 64 cubic inches, you can use the cube root to find the side length, which is 4 inches.
Engineering: Engineers use cube roots in various calculations, such as determining the dimensions of structures and designing machines.
Physics: Cube roots appear in some physics formulas, such as those related to volume and density.
Finance: In finance, cube roots can be used in calculations involving growth rates and compound interest.

So, while it might seem like a purely academic concept, cube roots do have real-world relevance.

Conclusion

Alright, that wraps up our discussion on finding the cube root of 64/27. Remember, the key is to break the problem down into smaller, manageable steps. By finding the cube root of the numerator and the denominator separately, you can easily solve the problem. And with a little practice, you'll be finding cube roots like a pro! Keep practicing, and don't be afraid to ask for help if you get stuck. You got this! Happy calculating!