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
00:00 Simplify the following problem
00:04 According to the laws of exponents, a fraction raised to a negative power (-N)
00:08 is equal to the reciprocal fraction raised to the opposite power(N)
00:12 We will apply this formula to our exercise
00:15 We'll convert to the reciprocal number and raise to the opposite power
00:25 According to the laws of exponents, a fraction that is raised to the power (N)
00:28 is equal to a fraction where both the numerator and denominator are raised to the power (N)
00:31 We will apply this formula to our exercise
00:35 We'll raise both the numerator and denominator to the appropriate power, maintaining the parentheses
00:41 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:

(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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