Evaluate (10/17)^(-5): Negative Exponent Fraction Problem

Negative Exponents with Fraction Bases

Insert the corresponding expression:

(1017)5= \left(\frac{10}{17}\right)^{-5}=

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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 the power (-N)
00:07 is equal to its reciprocal raised to the opposite power (N)
00:10 We'll apply this formula to our exercise
00:14 Let's invert the fraction
00:18 and raise it to the opposite power (times(-1))
00:21 That's the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Insert the corresponding expression:

(1017)5= \left(\frac{10}{17}\right)^{-5}=

2

Step-by-step solution

To solve the problem, we will follow these steps:

  • Step 1: Identify the expression and apply the rule for negative exponents.
  • Step 2: Take the reciprocal of the given fraction 1017 \frac{10}{17} .
  • Step 3: Raise the reciprocal to the positive power of 5.

Now, let's work through each step:
Step 1: We start with the problem expression (1017)5 \left(\frac{10}{17}\right)^{-5} . According to the laws of exponents, a negative exponent means the reciprocal of the base should be raised to the positive of that exponent.
Step 2: Take the reciprocal of 1017 \frac{10}{17} , which is 1710 \frac{17}{10} .
Step 3: Raise the reciprocal 1710 \frac{17}{10} to the power of 5, resulting in (1710)5 \left(\frac{17}{10}\right)^5 .

Therefore, the equivalent expression is (1710)5 \left(\frac{17}{10}\right)^5 .

3

Final Answer

(1710)5 \left(\frac{17}{10}\right)^5

Key Points to Remember

Essential concepts to master this topic
  • Rule: Negative exponent means reciprocal with positive exponent
  • Technique: (1017)5=(1710)5 \left(\frac{10}{17}\right)^{-5} = \left(\frac{17}{10}\right)^{5}
  • Check: Flip fraction first, then apply positive power ✓

Common Mistakes

Avoid these frequent errors
  • Applying negative sign to numerator only
    Don't make just the numerator negative like (1017)5 \left(\frac{-10}{17}\right)^{5} = completely wrong answer! The negative exponent affects the entire base structure, not individual numbers. Always flip the entire fraction first, then apply the positive exponent.

Practice Quiz

Test your knowledge with interactive questions

\( \)Choose the corresponding expression:

\( \left(\frac{1}{2}\right)^2= \)

FAQ

Everything you need to know about this question

Why does a negative exponent mean flipping the fraction?

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A negative exponent means "one over" the positive power. So an=1an a^{-n} = \frac{1}{a^n} . When your base is already a fraction, taking the reciprocal flips it!

Do I need to calculate the actual numerical value?

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Not usually! Most problems ask for the equivalent expression, which means keeping it as (1710)5 \left(\frac{17}{10}\right)^{5} rather than computing the huge number.

What if the exponent was positive instead of negative?

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Then you'd keep the fraction as-is! (1017)5 \left(\frac{10}{17}\right)^{5} stays exactly that way. Only negative exponents require flipping the fraction.

Can I flip the fraction and keep the negative exponent?

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No! Once you flip the fraction, the exponent becomes positive. (1017)5=(1710)5 \left(\frac{10}{17}\right)^{-5} = \left(\frac{17}{10}\right)^{5} , not (1710)5 \left(\frac{17}{10}\right)^{-5} .

How do I remember which way to flip?

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Think "opposite"! If the original fraction is smaller than 1 (like 10/17), flipping makes it larger than 1 (17/10). The negative exponent transforms to its opposite behavior.

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