Calculate (9/20) Raised to the Sixth Power: Fraction Exponent Problem

Q: Why does the exponent go to both the top and bottom of the fraction?

Because $ \left(\frac{9}{20}\right)^6 $ means multiplying the entire fraction by itself 6 times! Each multiplication affects both numerator and denominator, so the exponent distributes to both parts.

Q: What's the difference between this and just multiplying by 6?

Huge difference! $ \left(\frac{9}{20}\right)^6 $ means repeated multiplication, while $ \frac{9 \times 6}{20 \times 6} $ is just scaling. The power creates much larger numbers!

Q: Do I need to calculate the actual values of 9⁶ and 20⁶?

Not necessarily! The question asks for the expression, not the final number. $ \frac{9^6}{20^6} $ is the correct mathematical form.

Q: How can I remember this rule?

Think of it as "power distributes" - when you raise a fraction to a power, that power must go to every part of the fraction. The exponent can't skip the denominator!

Q: What if the fraction had addition or subtraction in it?

Be careful! This rule only works for multiplication and division (fractions). For expressions like $ (a + b)^n $, you need different techniques like the binomial theorem.

Fraction Exponents with Power Distribution Rules

Insert the corresponding expression:

$\left(\frac{9}{20}\right)^6=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

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, each raised to the same power (N)

00:12 We will apply this formula to our exercise

00:15 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:

$\left(\frac{9}{20}\right)^6=$

Step-by-step solution

To solve this problem, we will use the exponent rule for fractions, which states that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

Step 1: Recognize that the given expression is $\left(\frac{9}{20}\right)^6$ . This means we have a fraction base with an exponent 6.
Step 2: Apply the exponent rule: $\left(\frac{9}{20}\right)^6 = \frac{9^6}{20^6}$ .
Step 3: Compare this with the given multiple choices to select the correct one.

Upon comparing, we see that the correct choice from the given options is $\frac{9^6}{20^6}$ , which matches the expression derived using the exponent rule.

Therefore, the solution to the problem is $\frac{9^6}{20^6}$ .

Final Answer

$\frac{9^6}{20^6}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a fraction to a power, apply exponent to both numerator and denominator
Technique: $\left(\frac{9}{20}\right)^6 = \frac{9^6}{20^6}$ distributes the 6 to both parts
Check: Verify the exponent appears on both top and bottom numbers ✓

Common Mistakes

Avoid these frequent errors

Applying the exponent to only the numerator or denominator
Don't write $\left(\frac{9}{20}\right)^6 = \frac{9^6}{20}$ or $\frac{9}{20^6}$ = wrong fraction! The exponent must distribute to BOTH parts. Always apply the power rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why does the exponent go to both the top and bottom of the fraction?

Because $\left(\frac{9}{20}\right)^6$ means multiplying the entire fraction by itself 6 times! Each multiplication affects both numerator and denominator, so the exponent distributes to both parts.

What's the difference between this and just multiplying by 6?

Huge difference! $\left(\frac{9}{20}\right)^6$ means repeated multiplication, while $\frac{9 \times 6}{20 \times 6}$ is just scaling. The power creates much larger numbers!

Do I need to calculate the actual values of 9⁶ and 20⁶?

Not necessarily! The question asks for the expression, not the final number. $\frac{9^6}{20^6}$ is the correct mathematical form.

How can I remember this rule?

Think of it as "power distributes" - when you raise a fraction to a power, that power must go to every part of the fraction. The exponent can't skip the denominator!

What if the fraction had addition or subtraction in it?

Be careful! This rule only works for multiplication and division (fractions). For expressions like $(a + b)^n$ , you need different techniques like the binomial theorem.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime