Calculate (3/2)³: Evaluating the Cube of a Fraction

Q: Why do I raise both parts of the fraction to the power?

When you have $ \left(\frac{a}{b}\right)^n $, it means multiply the entire fraction by itself n times. This is equivalent to $ \frac{a^n}{b^n} $, so both parts get the exponent!

Q: Can I simplify the fraction before or after applying the exponent?

You can do either! $ \frac{3}{2} $ is already in simplest form, so it doesn't matter here. But sometimes simplifying first makes the calculations easier with smaller numbers.

Q: What if the exponent is negative?

A negative exponent means flip the fraction and make the exponent positive! For example: $ \left(\frac{3}{2}\right)^{-3} = \left(\frac{2}{3}\right)^3 = \frac{8}{27} $.

Q: How do I remember which number goes where?

Top stays top, bottom stays bottom - just with exponents applied! The numerator becomes $ 3^3 $ and denominator becomes $ 2^3 $.

Exponent Rules with Fractional Bases

Insert the corresponding expression:

$\left(\frac{3}{2}\right)^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, a fraction raised to a power (N)

00:08 equals the numerator and denominator raised to the same power (N)

00:11 We will apply this formula to our exercise

00:17 We'll calculate each power and substitute accordingly

00:36 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{3}{2}\right)^3=$

Step-by-step solution

To solve $\left(\frac{3}{2}\right)^3$ , follow these steps:

Step 1: Identify the fraction to be raised to a power, $\frac{3}{2}$ , and the exponent, 3.
Step 2: Apply the power to both the numerator and the denominator separately using the exponent rule for fractions.

Let's evaluate:

Raise the numerator 3 to the power of 3:

$3^3 = 3 \times 3 \times 3 = 27$

Raise the denominator 2 to the power of 3:

$2^3 = 2 \times 2 \times 2 = 8$

Combine these results into the fraction $\frac{27}{8}$ .

Therefore, $\left(\frac{3}{2}\right)^3 = \frac{27}{8}$ .

Given several answer choices, the correct choice is 4: $\frac{27}{8}$ .

Final Answer

$\frac{27}{8}$

Key Points to Remember

Essential concepts to master this topic

Rule: Raise both numerator and denominator to the given power
Technique: Calculate $3^3 = 27$ and $2^3 = 8$ separately
Check: Verify $\frac{27}{8}$ cannot be simplified further ✓

Common Mistakes

Avoid these frequent errors

Only raising the numerator to the power
Don't calculate just $3^3 = 27$ and keep denominator as 2 = $\frac{27}{2}$ ! This ignores the exponent rule for fractions and gives a completely wrong result. Always raise both the numerator AND denominator to the same power.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I raise both parts of the fraction to the power?

When you have $\left(\frac{a}{b}\right)^n$ , it means multiply the entire fraction by itself n times. This is equivalent to $\frac{a^n}{b^n}$ , so both parts get the exponent!

Can I simplify the fraction before or after applying the exponent?

You can do either! $\frac{3}{2}$ is already in simplest form, so it doesn't matter here. But sometimes simplifying first makes the calculations easier with smaller numbers.

What if the exponent is negative?

A negative exponent means flip the fraction and make the exponent positive! For example: $\left(\frac{3}{2}\right)^{-3} = \left(\frac{2}{3}\right)^3 = \frac{8}{27}$ .

How do I remember which number goes where?

Top stays top, bottom stays bottom - just with exponents applied! The numerator becomes $3^3$ and denominator becomes $2^3$ .

Is there a shortcut for cubing fractions?

Not really - you need to cube both parts separately. But you can use patterns: cubing means multiplying by itself three times, so $3^3 = 3 \times 3 \times 3 = 27$ .

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