Evaluate (6×8)/(2×7) Raised to the Power of -5: Complex Fraction Exercise

Q: Why does a negative exponent flip the fraction?

A negative exponent means "take the reciprocal". Since $ a^{-n} = \frac{1}{a^n} $, when you have $ \left(\frac{6×8}{2×7}\right)^{-5} $, it becomes $ \frac{1}{\left(\frac{6×8}{2×7}\right)^5} $, which simplifies to the flipped fraction with positive exponent.

Q: Do I need to calculate the actual numbers first?

No! You can apply the negative exponent rule directly to the fraction form. Converting $ \left(\frac{6×8}{2×7}\right)^{-5} $ to $ \left(\frac{2×7}{6×8}\right)^5 $ is correct without calculating 48/14 first.

Q: Does the negative exponent make the whole answer negative?

No! A negative exponent only affects the position of terms (flipping fractions), not the sign of the final result. The answer $ \left(\frac{2×7}{6×8}\right)^5 $ is still positive.

Q: What if I see a negative sign in front of the parentheses?

Be careful to distinguish between a negative exponent and a negative coefficient. In this problem, the -5 is an exponent, not a negative sign multiplying the entire expression.

Q: How can I remember the negative exponent rule?

Think: "Negative exponent = flip and make positive". Just like $ 2^{-3} = \frac{1}{2^3} $, fractions with negative exponents flip: $ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n $.

Negative Exponents with Complex Fractions

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

00:09 Let's simplify this problem together.

00:13 When a fraction is raised to a negative exponent, like negative N, just flip the fraction and use positive N.

00:19 So, it's the reciprocal fraction raised to the positive exponent N.

00:24 Alright, let's use this rule in our exercise.

00:28 We'll change to the reciprocal and raise it to the positive exponent.

00:32 And there you go! That's the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

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

Step-by-step solution

To solve the problem, we need to find the equivalent expression for the given negative exponent:

Step 1: Identify the base fraction as $\frac{6\times8}{2\times7}$ .

Step 2: Apply the negative exponent rule:

The expression $\left(\frac{6\times8}{2\times7}\right)^{-5}$ can be rewritten using the property that $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ .

Thus, we have:

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

Therefore, the correct equivalent expression is:

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

Hence, the choice corresponding to this expression is correct.

Therefore, the solution to the problem is $\left(\frac{2\times7}{6\times8}\right)^5$ , which corresponds to choice 1.

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent flips the fraction and makes exponent positive
Technique: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ switches numerator and denominator
Check: Verify by converting back: $\left(\frac{2×7}{6×8}\right)^5$ raised to -1/5 gives original ✓

Common Mistakes

Avoid these frequent errors

Making the exponent negative without flipping the fraction
Don't write $-\left(\frac{6×8}{2×7}\right)^5$ = wrong negative result! The negative exponent rule doesn't add a negative sign to the front. Always flip the fraction first, then make 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". Since $a^{-n} = \frac{1}{a^n}$ , when you have $\left(\frac{6×8}{2×7}\right)^{-5}$ , it becomes $\frac{1}{\left(\frac{6×8}{2×7}\right)^5}$ , which simplifies to the flipped fraction with positive exponent.

Do I need to calculate the actual numbers first?

No! You can apply the negative exponent rule directly to the fraction form. Converting $\left(\frac{6×8}{2×7}\right)^{-5}$ to $\left(\frac{2×7}{6×8}\right)^5$ is correct without calculating 48/14 first.

Does the negative exponent make the whole answer negative?

No! A negative exponent only affects the position of terms (flipping fractions), not the sign of the final result. The answer $\left(\frac{2×7}{6×8}\right)^5$ is still positive.

What if I see a negative sign in front of the parentheses?

Be careful to distinguish between a negative exponent and a negative coefficient. In this problem, the -5 is an exponent, not a negative sign multiplying the entire expression.

How can I remember the negative exponent rule?

Think: "Negative exponent = flip and make positive". Just like $2^{-3} = \frac{1}{2^3}$ , fractions with negative exponents flip: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ .

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