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Chapter 4: Problem 2

Write each expression using a positive exponent. $$(-7)^{-1}$$

Short Answer

Expert verified
\((-7)^{-1}\) equals \(\frac{1}{-7}\).

Step by step solution

01

Understanding Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. That means for any nonzero number \(a\), \(a^{-n} = \frac{1}{a^n}\). In this exercise, we have the expression \((-7)^{-1}\).
02

Applying the Definition of Negative Exponents

According to the definition, replace \((-7)^{-1}\) with the reciprocal of \(-7\) raised to the positive exponent 1. This gives us \((-7)^{-1} = \frac{1}{(-7)^1}\).
03

Simplifying the Expression

Now simplify the expression \(\frac{1}{(-7)^1}\). Since \((-7)^1 = -7\), this simplifies to \(\frac{1}{-7}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal
When we talk about the reciprocal, we are referring to flipping a fraction or, in other words, taking the inverse of a number.
For example, the reciprocal of a number like 2 is \( \frac{1}{2} \). When handling negative exponents, understanding reciprocals becomes very important.
In this context, a negative exponent tells us to find the reciprocal of the base. For instance, if we have \( a^{-n} \), it translates to \( \frac{1}{a^n} \).
Remember:
  • Reciprocal flips a number: if it's \( x \), it becomes \( \frac{1}{x} \).
  • Works for both positive and negative numbers.
  • Aids in converting negative exponents to positive ones.
When you encounter negative exponents, always think about finding the reciprocal as the first step. This will help you transition to using positive exponents.
Positive Exponents
Positive exponents reflect the multiplication of a number by itself a certain number of times.
For example, \( 3^2 \) means that 3 is multiplied by itself, resulting in 9.
Positive exponents are more straightforward because you are repeatedly multiplying the base number. Some key points include:
  • Always deal with multiplication.
  • Simplifies expressions into standard numbers.
  • No reciprocal needed if the exponent is already positive.
In the solution of the given exercise, after converting the negative exponent to a reciprocal, we work with positive exponents. This means we just need to consider the number \( (-7)^1 \) which is straightforward as it equals \( -7 \).
Positive exponents lead to a more direct computation, unlike negative exponents, which require initial transformation.
Exponent Rules
Exponent rules are vital in interpreting and simplifying expressions, especially when dealing with both negative and positive exponents.
Here's a quick rundown of essential rules to remember:
  • Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
  • Power of a Power: \( (a^m)^n = a^{m\cdot n} \)
  • Negative Exponent: \( a^{-n} = \frac{1}{a^n} \) - used to shift to positive exponents.
  • Zero Exponent: \( a^0 = 1 \) provided that \( a \) is not zero.
These rules allow us to transform, simplify, and solve various problems involving exponents. In particular, when handling negative exponents like \( (-7)^{-1} \), the rule \( a^{-n} = \frac{1}{a^n} \) applies perfectly to convert them into an expression with positive exponents, making computations straightforward and answers clearer.

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Most popular questions from this chapter

Evaluate each expression if \(s=-2\) and \(t=3\). $$t^{-4}$$

For each expression, use parentheses to group the numbers together and to group the powers with like bases together. Example: \(a \cdot 4 \cdot a^{3} \cdot 2=(4 \cdot 2)\left(a \cdot a^{3}\right)\) $$3 \cdot x^{4} \cdot(-5) \cdot x^{2}$$

Determine whether each number is prime or composite. $$99$$

Find each equivalent measure. $$1 \mathrm{yd}=\underline{?} \text { in. }$$

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