Calculate (6×8)/(2×7) Raised to the Negative 5th Power

Q: Why does a negative exponent flip the fraction?

A negative exponent means "take the reciprocal". So $ \left(\frac{6×8}{2×7}\right)^{-5} $ becomes $ \left(\frac{2×7}{6×8}\right)^{5} $ - we flip the fraction and make the exponent positive!

Q: Can I distribute the exponent to each number separately?

Yes! $ \left(\frac{6×8}{2×7}\right)^{-5} = \frac{6^{-5}×8^{-5}}{2^{-5}×7^{-5}} $ is valid using the power rule. This gives you another correct way to express the same answer.

Q: How do I know which answer form to choose?

Look for equivalent expressions! Different forms can be mathematically equal. The key is understanding that negative exponents flip fractions and can be distributed to factors using exponent rules.

Negative Exponents with Fraction Bases

Insert the corresponding expression:

$\left(\frac{6\times8}{2\times7}\right)^{-5}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's simplify this math problem, shall we?

00:14 According to exponent rules, a fraction to the power of N

00:18 means both the numerator and denominator are raised to power N.

00:23 We'll use this rule for our exercise.

00:26 We'll raise both the top and bottom numbers to the power of N.

00:36 Also, when a product is raised to the power of N,

00:40 each number in the product is raised independently to power N.

00:44 Let's apply this to solve our problem.

00:54 And that's the solution! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{6\times8}{2\times7}\right)^{-5}=$

Final Answer

A'+C' are correct

Key Points to Remember

Essential concepts to master this topic

Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$ flips the base to denominator
Power of Quotient: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ flips fraction and makes exponent positive
Verify: Check that $\left(\frac{12}{14}\right)^{-5} = \left(\frac{14}{12}\right)^5$ produces same result ✓

Common Mistakes

Avoid these frequent errors

Applying negative exponent only to numerator or denominator
Don't write $\left(\frac{6×8}{2×7}\right)^{-5} = \frac{(6×8)^{-5}}{(2×7)^5}$ = wrong answer! This misapplies the exponent rule and creates incorrect expressions. Always apply the negative exponent to the entire fraction by flipping it and making the exponent positive.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does a negative exponent flip the fraction?

A negative exponent means "take the reciprocal". So $\left(\frac{6×8}{2×7}\right)^{-5}$ becomes $\left(\frac{2×7}{6×8}\right)^{5}$ - we flip the fraction and make the exponent positive!

Do I need to calculate 6×8 and 2×7 first?

Not necessarily! You can work with the expression symbolically. Whether you simplify $\frac{6×8}{2×7} = \frac{48}{14} = \frac{24}{7}$ first or keep it as separate factors depends on what the problem asks for.

What's the difference between the answer choices?

Choice A applies the negative exponent incorrectly to both parts. Choice B mixes positive and negative exponents wrong. Choice C distributes the exponent to individual factors. Multiple correct forms exist - that's why "A'+C' are correct" is the answer!

Can I distribute the exponent to each number separately?

Yes! $\left(\frac{6×8}{2×7}\right)^{-5} = \frac{6^{-5}×8^{-5}}{2^{-5}×7^{-5}}$ is valid using the power rule. This gives you another correct way to express the same answer.

How do I know which answer form to choose?

Look for equivalent expressions! Different forms can be mathematically equal. The key is understanding that negative exponents flip fractions and can be distributed to factors using exponent rules.

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