Calculate (5×9÷20)³: Cube of a Fraction Problem

Q: Why does the exponent apply to both parts of the fraction?

When you have $ \left(\frac{a}{b}\right)^3 $, you're multiplying the entire fraction by itself 3 times. This means both the top and bottom get cubed: $ \frac{a}{b} \times \frac{a}{b} \times \frac{a}{b} = \frac{a^3}{b^3} $.

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

Yes! You could first calculate $ \frac{5\times9}{20} = \frac{45}{20} = \frac{9}{4} $, then cube it: $ \left(\frac{9}{4}\right)^3 = \frac{9^3}{4^3} $. Both methods give the same answer!

Q: How do I expand the product in the numerator?

Use the power of a product rule: $ (5\times9)^3 = 5^3\times9^3 $. This is because when you multiply factors raised to the same power, you can separate them.

Q: What's the difference between the correct and incorrect answers?

The correct answer $ \frac{5^3\times9^3}{20^3} $ applies the cube to everything. Wrong answers like $ \frac{5^3\times9^3}{20} $ forget to cube the denominator.

Q: Do I need to calculate the final numerical answer?

Not necessarily! Leaving it as $ \frac{5^3\times9^3}{20^3} $ shows you understand the exponent rules. You could calculate: $ \frac{125\times729}{8000} = \frac{91125}{8000} $ if needed.

Exponent Rules with Fraction Multiplication

$\left(\frac{5\times9}{20}\right)^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's simplify this problem together.

00:08 Remember, when a fraction is raised to the power of N, pause, each part, both the numerator and denominator, is also raised to N.

00:16 So, we will use this rule for our exercise.

00:20 Notice the numerator is a product. Be careful with the parentheses.

00:25 Recall, if a product is raised to the power of N, pause, each factor is raised to N.

00:32 Let's apply this formula to solve our exercise.

00:35 And that's how we find the solution. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\left(\frac{5\times9}{20}\right)^3=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Evaluate the product in the numerator.
Step 2: Apply the power of a fraction rule.
Step 3: Simplify using the power of a product rule.

Now, let's work through each step:

Apply the power of a fraction rule:
$\frac{45^3}{20^3} = \frac{(5 \times 9)^3}{20^3} = \frac{5^3 \times 9^3}{20^3}$ .

Comparing this with the given choices, we find that:

The correct answer is Choice 1: $\frac{5^3 \times 9^3}{20^3}$ .

Final Answer

$\frac{5^3\times9^3}{20^3}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to entire fraction: numerator and denominator
Technique: $\left(\frac{5\times9}{20}\right)^3 = \frac{(5\times9)^3}{20^3}$
Check: Verify using product rule: $(5\times9)^3 = 5^3\times9^3$ ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to the numerator
Don't cube just the numerator like $\frac{(5\times9)^3}{20}$ = wrong answer! The exponent applies to the entire fraction. Always apply the power to both the numerator AND denominator: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does the exponent apply to both parts of the fraction?

When you have $\left(\frac{a}{b}\right)^3$ , you're multiplying the entire fraction by itself 3 times. This means both the top and bottom get cubed: $\frac{a}{b} \times \frac{a}{b} \times \frac{a}{b} = \frac{a^3}{b^3}$ .

Can I simplify the fraction before applying the exponent?

Yes! You could first calculate $\frac{5\times9}{20} = \frac{45}{20} = \frac{9}{4}$ , then cube it: $\left(\frac{9}{4}\right)^3 = \frac{9^3}{4^3}$ . Both methods give the same answer!

How do I expand the product in the numerator?

Use the power of a product rule: $(5\times9)^3 = 5^3\times9^3$ . This is because when you multiply factors raised to the same power, you can separate them.

What's the difference between the correct and incorrect answers?

The correct answer $\frac{5^3\times9^3}{20^3}$ applies the cube to everything. Wrong answers like $\frac{5^3\times9^3}{20}$ forget to cube the denominator.

Do I need to calculate the final numerical answer?

Not necessarily! Leaving it as $\frac{5^3\times9^3}{20^3}$ shows you understand the exponent rules. You could calculate: $\frac{125\times729}{8000} = \frac{91125}{8000}$ if needed.

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