Solve (3×7)/(5×8) Raised to Power -3: Negative Exponent Challenge

Negative Exponents with Fraction Bases

Insert the corresponding expression:

(3×75×8)−3= \left(\frac{3\times7}{5\times8}\right)^{-3}=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:11 Let's simplify this problem together.
00:15 When a fraction is raised to a negative power, like negative N, remember this rule:
00:22 It becomes the reciprocal of the fraction raised to the positive power, N. Pretty cool, right?
00:28 Now, we'll try this rule out in our exercise.
00:33 We'll switch to the reciprocal fraction and raise it to the opposite power. Let's give it a try!
00:39 A fraction raised to a power, N, means both parts are affected.
00:44 That means the top and bottom numbers are each raised to the power, N.
00:50 Let's apply this idea to our exercise now.
00:54 Remember to raise both the top and bottom while keeping everything in parentheses.
01:00 And there you go! That's how you solve it.

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Insert the corresponding expression:

(3×75×8)−3= \left(\frac{3\times7}{5\times8}\right)^{-3}=

2

Step-by-step solution

To simplify the expression (3×75×8)−3 \left(\frac{3\times7}{5\times8}\right)^{-3} , we follow these steps:

  • Step 1: Apply the rule for negative exponents, which states that a−b=1ab a^{-b} = \frac{1}{a^b} . For fractions, (ab)−n=bnan \left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n} .
  • Step 2: Rewrite the expression by applying this rule:
    (3×75×8)−3=(5×8)3(3×7)3 \left(\frac{3\times7}{5\times8}\right)^{-3} = \frac{(5\times8)^3}{(3\times7)^3}
  • Step 3: Simplify the expression by recognizing the bases to the power of 3: (5×8)3(3×7)3 \frac{(5\times8)^3}{(3\times7)^3}

The expression can be left as is because it matches one of the choices provided. Performing further calculations here is unnecessary since we have correctly applied the negative exponent rule.

Therefore, the correct simplified form of the expression is (5×8)3(3×7)3 \frac{(5\times8)^3}{(3\times7)^3} , which corresponds to choice 2.

3

Final Answer

(5×8)3(3×7)3 \frac{\left(5\times8\right)^3}{\left(3\times7\right)^3}

Key Points to Remember

Essential concepts to master this topic
  • Rule: For fractions, (ab)−n=bnan \left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n} (flip and make positive)
  • Technique: (3×75×8)−3=(5×8)3(3×7)3 \left(\frac{3×7}{5×8}\right)^{-3} = \frac{(5×8)^3}{(3×7)^3} by flipping the fraction
  • Check: Verify numerator and denominator switched positions and exponent became positive ✓

Common Mistakes

Avoid these frequent errors
  • Distributing negative exponent incorrectly to individual terms
    Don't write 3−3×7−353×83 \frac{3^{-3}×7^{-3}}{5^3×8^3} = mixed positive and negative exponents! This misapplies the negative exponent rule and creates an incorrect expression. Always flip the entire fraction first, then apply the positive exponent to both numerator and denominator.

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?

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A negative exponent means "take the reciprocal" - so (ab)−n \left(\frac{a}{b}\right)^{-n} becomes (ba)n \left(\frac{b}{a}\right)^{n} . It's like saying "flip it upside down and make the exponent positive!"

Do I need to calculate 3×7 and 5×8 first?

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No! Keep the products as (3×7) and (5×8) in your answer. The question asks for the equivalent expression, not the numerical result.

What if I see 53×833−3×7−3 \frac{5^3×8^3}{3^{-3}×7^{-3}} as an answer choice?

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This is incorrect because it mixes positive and negative exponents. When you flip a fraction with a negative exponent, all terms in both numerator and denominator get positive exponents.

How do I remember the negative exponent rule for fractions?

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Think: "Negative means flip!" When you see a negative exponent on a fraction, flip the fraction upside down and make the exponent positive. Practice with simple examples like (12)−1=21 \left(\frac{1}{2}\right)^{-1} = \frac{2}{1} .

Can I work with the individual factors instead?

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While you could distribute the exponent to get 53×8333×73 \frac{5^3×8^3}{3^3×7^3} , the given answer choices use grouped factors like (5×8)3 (5×8)^3 . Match the format requested!

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