Simplify x^-a: Understanding Negative Exponent Notation

Negative Exponents with Reciprocal Rules

x−a=? x^{-a}=\text{?}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Simplify the following expression
00:03 According to the laws of exponents, any number(A) raised to the power of(N)
00:07 equals 1 divided by the number(A) raised to the power of(-N)
00:10 Let's apply this to the question
00:13 The number(X) becomes 1 divided by X
00:16 And the exponent(-A) becomes -(-A)
00:19 A negative multiplied by a negative becomes a positive hence the power is A
00:22 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

x−a=? x^{-a}=\text{?}

2

Step-by-step solution

We use the exponential property of a negative exponent:

b−n=1bn b^{-n}=\frac{1}{b^n} We apply it to the problem:

x−a=1xa x^{-a}=\frac{1}{x^a} Therefore, the correct answer is option C.

3

Final Answer

1xa \frac{1}{x^a}

Key Points to Remember

Essential concepts to master this topic
  • Rule: Negative exponents create reciprocals: b−n=1bn b^{-n} = \frac{1}{b^n}
  • Technique: Move base to denominator and make exponent positive: x−a=1xa x^{-a} = \frac{1}{x^a}
  • Check: Verify by multiplying result by original: x−a⋅xa=1 x^{-a} \cdot x^a = 1 ✓

Common Mistakes

Avoid these frequent errors
  • Adding negative sign to the base instead of using reciprocal rule
    Don't write x−a=−xa x^{-a} = -x^a = wrong negative placement! The negative exponent doesn't make the result negative, it creates a reciprocal. Always use the reciprocal rule: x−a=1xa x^{-a} = \frac{1}{x^a} .

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 create a fraction?

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Think of it as division instead of multiplication. A positive exponent means "multiply x by itself a times," while a negative exponent means "divide 1 by x multiplied by itself a times."

Is x−a x^{-a} the same as −xa -x^a ?

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No! x−a=1xa x^{-a} = \frac{1}{x^a} (always positive if x is positive), while −xa -x^a is the negative of xa x^a . The negative sign affects completely different things!

What if the base is a fraction like (23)−2 (\frac{2}{3})^{-2} ?

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Use the same rule! (23)−2=1(23)2=149=94 (\frac{2}{3})^{-2} = \frac{1}{(\frac{2}{3})^2} = \frac{1}{\frac{4}{9}} = \frac{9}{4} . You can also flip the fraction first: (23)−2=(32)2=94 (\frac{2}{3})^{-2} = (\frac{3}{2})^2 = \frac{9}{4} .

Can negative exponents make numbers bigger?

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Yes! When the base is between 0 and 1, negative exponents create numbers greater than 1. For example: (0.5)−2=10.25=4 (0.5)^{-2} = \frac{1}{0.25} = 4 !

How do I remember the reciprocal rule?

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Remember: "Negative exponent = flip it!" The negative sign tells you to take the reciprocal (flip the fraction). x−a x^{-a} becomes 1xa \frac{1}{x^a} .

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