Calculate (4/5)³: Evaluating the Cube of a Fraction

Q: Is there a shortcut for cubing fractions?

Yes! Use the rule: $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $. Just cube the top number and cube the bottom number separately, then write as a new fraction.

Q: How do I know if my final fraction can be simplified?

Check if the numerator and denominator share any common factors. For $ \frac{64}{125} $, since 64 = 2⁶ and 125 = 5³, they share no common factors, so it's already simplified.

Q: What if I get confused with the multiplication?

Write it out step by step! For $ 4^3 $, write 4 × 4 × 4 = 16 × 4 = 64. For $ 5^3 $, write 5 × 5 × 5 = 25 × 5 = 125.

Q: Can I use a calculator for this?

Absolutely! But make sure you understand the process. Calculate $ 4^3 $ and $ 5^3 $ separately, then form the fraction $ \frac{64}{125} $.

Exponent Rules with Fractional Bases

Insert the corresponding expression:

$\left(\frac{4}{5}\right)^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's simplify this problem together.

00:09 Remember, the law of exponents states that when a fraction is raised to a power like N,

00:14 both the numerator and the denominator need to be raised to the power of N.

00:20 We'll apply this rule to our exercise now.

00:23 First, calculate each power and then substitute the values in.

00:45 And that's how you find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{4}{5}\right)^3=$

Step-by-step solution

To solve this problem, we will evaluate the expression $\left(\frac{4}{5}\right)^3$ following these steps:

Step 1: Apply the power to the numerator.
Step 2: Apply the power to the denominator.
Step 3: Calculate the results for both and simplify if possible.

Let's go through each step:

Step 1: Calculate the cube of the numerator, which is 4. That is, $4^3 = 4 \times 4 \times 4 = 64$ .

Step 2: Calculate the cube of the denominator, which is 5. That is, $5^3 = 5 \times 5 \times 5 = 125$ .

Step 3: Combine these results into a fraction: $\frac{64}{125}$ .

Therefore, the expression $\left(\frac{4}{5}\right)^3$ simplifies to $\frac{64}{125}$ .

Upon comparing with the given choices, the correct answer is choice 1: $\frac{64}{125}$ .

Final Answer

$\frac{64}{125}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply the exponent to both numerator and denominator separately
Technique: Calculate $4^3 = 64$ and $5^3 = 125$ individually
Check: Verify $\frac{64}{125}$ cannot simplify further ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to the numerator
Don't calculate $\left(\frac{4}{5}\right)^3$ as $\frac{4^3}{5}$ = $\frac{64}{5}$ ! This ignores the denominator and gives a completely wrong answer. Always apply the exponent to both the numerator AND denominator separately.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I apply the exponent to both parts of the fraction?

When you raise a fraction to a power, you're multiplying the entire fraction by itself multiple times. So $\left(\frac{4}{5}\right)^3 = \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5}$ , which equals $\frac{4 \times 4 \times 4}{5 \times 5 \times 5}$ .

Is there a shortcut for cubing fractions?

Yes! Use the rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ . Just cube the top number and cube the bottom number separately, then write as a new fraction.

How do I know if my final fraction can be simplified?

Check if the numerator and denominator share any common factors. For $\frac{64}{125}$ , since 64 = 2⁶ and 125 = 5³, they share no common factors, so it's already simplified.

What if I get confused with the multiplication?

Write it out step by step! For $4^3$ , write 4 × 4 × 4 = 16 × 4 = 64. For $5^3$ , write 5 × 5 × 5 = 25 × 5 = 125.

Can I use a calculator for this?

Absolutely! But make sure you understand the process. Calculate $4^3$ and $5^3$ separately, then form the fraction $\frac{64}{125}$ .

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