Simplify the Expression: (6^5)/(13^5×4^5) Step by Step

Q: Why can I combine 13^5 × 4^5 into (13 × 4)^5?

This uses the product rule for exponents: $ a^m \times b^m = (a \times b)^m $. Since both 13 and 4 have the same exponent 5, you can factor it out!

Q: What's the difference between this and just canceling exponents?

You're not canceling - you're using the quotient rule! When numerator and denominator have the same exponent, the whole fraction can be written with that exponent outside.

Q: Do I need to calculate 13 × 4 = 52 in my final answer?

No! Leave it as $ \left(\frac{6}{13 \times 4}\right)^5 $ to show your work clearly. The question asks for the corresponding expression, not a numerical calculation.

Q: What if the exponents were different numbers?

Then you cannot use this rule! The quotient rule $ \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m $ only works when the exponents are exactly the same.

Q: How do I check if my simplified form is correct?

Expand your answer back to the original form. $ \left(\frac{6}{13 \times 4}\right)^5 = \frac{6^5}{(13 \times 4)^5} = \frac{6^5}{13^5 \times 4^5} $ - it matches!

Exponent Properties with Quotient Rule

Insert the corresponding expression:

$\frac{6^5}{13^5\times4^5}=$

❤️ 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

00:12 Let's simplify this problem.

00:15 When multiplying, each factor raised to power N is the same as the whole product at power N.

00:22 We'll use this rule with parentheses and an exponent.

00:27 Next, for fractions, when a fraction is at power N, both top and bottom parts are raised to power N.

00:35 Now, we'll convert our expression, putting it inside parentheses with an exponent.

00:45 And there it is. We've found the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{6^5}{13^5\times4^5}=$

Step-by-step solution

To solve this problem, we aim to rewrite the given expression using the properties of exponents. The expression we need to deal with is $\frac{6^5}{13^5 \times 4^5}$ .

We can simplify this using the formula for powers of fractions, which states that if two exponents are the same, we can treat the fraction as a whole raised to that exponent: $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ .

Applying this to the problem, considering the expression $\frac{6^5}{(13 \times 4)^5}$ all raised to the power of 5 can be rewritten, using our formula, as a single fraction raised to the same power:
$\left(\frac{6}{13 \times 4}\right)^5$ .

This simplifies the entire expression to a clear fraction raised to the power 5. Therefore, the corresponding expression to the original problem is:

$\left(\frac{6}{13 \times 4}\right)^5$ .

Final Answer

$\left(\frac{6}{13\times4}\right)^5$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same powers, use $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$
Technique: Combine denominators first: $13^5 \times 4^5 = (13 \times 4)^5$
Check: Expand back: $\left(\frac{6}{52}\right)^5 = \frac{6^5}{52^5} = \frac{6^5}{13^5 \times 4^5}$ ✓

Common Mistakes

Avoid these frequent errors

Separating exponents incorrectly
Don't write $\frac{6}{13 \times 4^5}$ or $\frac{6}{13^5 \times 4}$ = wrong distribution of powers! This ignores that ALL terms in denominator have the same exponent. Always recognize that $13^5 \times 4^5 = (13 \times 4)^5$ before applying quotient rule.

Practice Quiz

Test your knowledge with interactive questions

$$ Choose the corresponding expression:

$\left(\frac{1}{2}\right)^2=$

FAQ

Everything you need to know about this question

Why can I combine 13^5 × 4^5 into (13 × 4)^5?

This uses the product rule for exponents: $a^m \times b^m = (a \times b)^m$ . Since both 13 and 4 have the same exponent 5, you can factor it out!

What's the difference between this and just canceling exponents?

You're not canceling - you're using the quotient rule! When numerator and denominator have the same exponent, the whole fraction can be written with that exponent outside.

Do I need to calculate 13 × 4 = 52 in my final answer?

No! Leave it as $\left(\frac{6}{13 \times 4}\right)^5$ to show your work clearly. The question asks for the corresponding expression, not a numerical calculation.

What if the exponents were different numbers?

Then you cannot use this rule! The quotient rule $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ only works when the exponents are exactly the same.

How do I check if my simplified form is correct?

Expand your answer back to the original form. $\left(\frac{6}{13 \times 4}\right)^5 = \frac{6^5}{(13 \times 4)^5} = \frac{6^5}{13^5 \times 4^5}$ - it matches!

🌟 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