Solve (4×8)/(5×17×3) Raised to Power of -4: Complete Expression

Q: Why doesn't the negative exponent make the answer negative?

The negative exponent only affects the base, not the sign of the result! It means "take the reciprocal" - flip the fraction upside down and make the exponent positive. The final answer is still positive.

Q: Do I need to calculate the actual numbers inside the fraction?

Not necessarily! The question asks for the expression, so you can leave it as $ \left(\frac{5×17×3}{4×8}\right)^4 $. Only calculate if the problem specifically asks for a numerical answer.

Q: What if I have a positive number raised to a negative power?

Same rule applies! For any base a: $ a^{-n} = \frac{1}{a^n} $. With fractions, it's even easier because you just flip the fraction.

Q: Can I simplify the numbers before applying the negative exponent rule?

You could, but it's usually easier to apply the negative exponent rule first. $ 4×8 = 32 $ and $ 5×17×3 = 255 $, but keeping it factored often matches the answer choices better.

Negative Exponents with Complex Fractions

Insert the corresponding expression:

$\left(\frac{4\times8}{5\times17\times3}\right)^{-4}=$

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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 exponent laws, a fraction raised to a negative power(-N)

00:06 is equal to its reciprocal raised to the opposite exponent (N)

00:10 We'll apply this formula to our exercise

00:14 We'll convert to the reciprocal number and raise it to the opposite power

00:22 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:

$\left(\frac{4\times8}{5\times17\times3}\right)^{-4}=$

Step-by-step solution

To solve this problem, we'll use the rule for negative exponents. Given the expression:

$\left(\frac{4\times8}{5\times17\times3}\right)^{-4}$

Step 1: Apply the property of exponents $(a/b)^{-n} = (b/a)^n$ to change the negative exponent to a positive one. Therefore, we take the reciprocal of the fraction:

$\left(\frac{5\times17\times3}{4\times8}\right)^4$

Therefore, the rewritten expression is $\left(\frac{5\times17\times3}{4\times8}\right)^4$ .

Comparing this with the given answer choices, we find:

Choice 1: $\left(\frac{5\times17\times3}{4\times8}\right)^4$ - Correct

Thus, the solution to the problem is $\left(\frac{5\times17\times3}{4\times8}\right)^4$ .

Final Answer

$\left(\frac{5\times17\times3}{4\times8}\right)^4$

Key Points to Remember

Essential concepts to master this topic

Rule: $(a/b)^{-n} = (b/a)^n$ flips fraction and changes sign
Technique: $\left(\frac{32}{255}\right)^{-4} = \left(\frac{255}{32}\right)^4$ by reciprocal rule
Check: Verify denominator becomes numerator and exponent becomes positive ✓

Common Mistakes

Avoid these frequent errors

Making the entire expression negative when seeing negative exponent
Don't think $\left(\frac{4×8}{5×17×3}\right)^{-4} = -\left(\frac{5×17×3}{4×8}\right)^4$ ! Negative exponents don't make the result negative - they only flip fractions and change the exponent sign. Always remember: negative exponent = reciprocal with positive exponent.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why doesn't the negative exponent make the answer negative?

The negative exponent only affects the base, not the sign of the result! It means "take the reciprocal" - flip the fraction upside down and make the exponent positive. The final answer is still positive.

Do I need to calculate the actual numbers inside the fraction?

Not necessarily! The question asks for the expression, so you can leave it as $\left(\frac{5×17×3}{4×8}\right)^4$ . Only calculate if the problem specifically asks for a numerical answer.

What if I have a positive number raised to a negative power?

Same rule applies! For any base a: $a^{-n} = \frac{1}{a^n}$ . With fractions, it's even easier because you just flip the fraction.

How do I remember which way to flip the fraction?

Think of it this way: negative exponent means "opposite" - so the opposite of division is multiplication. What was on bottom goes on top, and what was on top goes on bottom!

Can I simplify the numbers before applying the negative exponent rule?

You could, but it's usually easier to apply the negative exponent rule first. $4×8 = 32$ and $5×17×3 = 255$ , but keeping it factored often matches the answer choices better.

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