Simplify the Complex Fraction: (2⁴×3⁵)/(7⁴×11⁵)

Q: Why can't I just add the exponents 4 + 5 = 9?

You can only add exponents when multiplying the same base. Since 2 and 3 are different bases, $ 2^4 \times 3^5 $ stays as is. The rule $ a^m \times a^n = a^{m+n} $ only works for identical bases!

Q: How do I know which terms to group together?

Look for matching exponents in numerator and denominator. Here, $ 2^4 $ and $ 7^4 $ both have exponent 4, while $ 3^5 $ and $ 11^5 $ both have exponent 5.

Q: What if the exponents don't match up perfectly?

When exponents don't match, you cannot use the $ \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m $ rule. The expression would need to stay in its original form or be simplified using other methods.

Q: Can I multiply out the numbers first instead?

You could calculate $ 2^4 = 16 $, $ 3^5 = 243 $, etc., but that creates much larger numbers! Using exponent properties keeps the expression simpler and shows the mathematical structure better.

Q: Is there only one correct way to write the final answer?

The mathematically equivalent forms are correct, but $ \left(\frac{2}{7}\right)^4 \times \left(\frac{3}{11}\right)^5 $ is the most simplified form using proper exponent properties. Other forms might be mathematically equal but less elegant.

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 the power (N)

00:08 is equal to a fraction where both the numerator and denominator are raised to the power (N)

00:11 We will apply this formula to our exercise, in the reverse direction

00:14 We'll break it into 2 separate fractions, place inside of parentheses and raise to the power

00:25 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Insert the corresponding expression:

$\frac{2^4\times3^5}{7^4\times11^5}=$

2

Step-by-step solution

To solve the problem, we will utilize the properties of exponents to express the given fraction properly.

The given expression is $\frac{2^4 \times 3^5}{7^4 \times 11^5}$ . Our goal is to rewrite this expression using properties of exponents.

Let's break it down into two separate expressions based on the properties of division of exponents:

$\frac{2^4}{7^4} = \left(\frac{2}{7}\right)^4$ because $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ .
$\frac{3^5}{11^5} = \left(\frac{3}{11}\right)^5$ for the same reasons as above.

By rewriting the original expression using these properties, we have:

$\frac{2^4 \times 3^5}{7^4 \times 11^5} = \left(\frac{2}{7}\right)^4 \times \left(\frac{3}{11}\right)^5$ .

Therefore, the expression can be rewritten as $\left(\frac{2}{7}\right)^4 \times \left(\frac{3}{11}\right)^5$ . This corresponds to choice 3 in the provided options.

The correct transformation of the expression is effectively represented and confirmed with the properties of exponents.

3

Final Answer

$\left(\frac{2}{7}\right)^4\times\left(\frac{3}{11}\right)^5$

Key Points to Remember

Essential concepts to master this topic

Rule: Use $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ to separate same exponents
Technique: Split $\frac{2^4}{7^4}$ and $\frac{3^5}{11^5}$ into separate fraction powers
Check: Verify $\left(\frac{2}{7}\right)^4 \times \left(\frac{3}{11}\right)^5$ equals original expression ✓

Common Mistakes

Avoid these frequent errors

Trying to combine terms with different exponents
Don't combine $2^4 \times 3^5$ into $(2 \times 3)^{4+5}$ = wrong power rules! You can only add exponents when multiplying same bases. Always separate terms with matching exponents first.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why can't I just add the exponents 4 + 5 = 9?

+

You can only add exponents when multiplying the same base. Since 2 and 3 are different bases, $2^4 \times 3^5$ stays as is. The rule $a^m \times a^n = a^{m+n}$ only works for identical bases!

How do I know which terms to group together?

+

Look for matching exponents in numerator and denominator. Here, $2^4$ and $7^4$ both have exponent 4, while $3^5$ and $11^5$ both have exponent 5.

What if the exponents don't match up perfectly?

+

When exponents don't match, you cannot use the $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ rule. The expression would need to stay in its original form or be simplified using other methods.

Can I multiply out the numbers first instead?

+

You could calculate $2^4 = 16$ , $3^5 = 243$ , etc., but that creates much larger numbers! Using exponent properties keeps the expression simpler and shows the mathematical structure better.

Is there only one correct way to write the final answer?

+

The mathematically equivalent forms are correct, but $\left(\frac{2}{7}\right)^4 \times \left(\frac{3}{11}\right)^5$ is the most simplified form using proper exponent properties. Other forms might be mathematically equal but less elegant.

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Simplify the Complex Fraction: (2⁴×3⁵)/(7⁴×11⁵)

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