Problem 132 Simplify each expression. (a) ... [FREE SOLUTION]

Chapter 5: Problem 132

Simplify each expression. (a) \(6 r^{-1}\) (b) \((6 r)^{-1}\) (c) \((-6 r)^{-1}\)

Short Answer

Expert verified

(a) \(\frac{6}{r}\), (b) \(\frac{1}{6r}\), (c) \(-\frac{1}{6r}\).

Step by step solution

Simplify part (a)

For the expression \(6 r^{-1}\), recall that any term with a negative exponent means that term is in the denominator. Therefore, \(r^{-1} = \frac{1}{r}\). Now multiply by 6: \(6 r^{-1} = 6 \cdot \frac{1}{r} = \frac{6}{r}\).

Simplify part (b)

For the expression \((6 r)^{-1}\), again, a negative exponent means we take the reciprocal of the base. So, \((6 r)^{-1} = \frac{1}{6 r}\).

Simplify part (c)

For the expression \((-6 r)^{-1}\), similar to the previous step, take the reciprocal of the base. Therefore, \((-6 r)^{-1} = \frac{1}{-6 r} = -\frac{1}{6 r}\).

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Key Concepts

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

Negative Exponents

Negative exponents can be confusing at first, but they follow a simple rule. When you see a negative exponent, such as in the expression \(a^{-n}\), it means that you need to take the reciprocal of the base and change the exponent to positive. For example, \(x^{-3}\) becomes \(\frac{1}{x^3}\). This is derived from the fact that any number raised to an exponent represents repeated multiplication or division. Remembering this rule helps simplify expressions involving negative exponents.

Reciprocal

A reciprocal is essentially flipping a number or expression around. For example, the reciprocal of \(5\) is \(\frac{1}{5}\) and vice versa. In the context of algebra, finding the reciprocal of a product like \(ab\) means creating an expression like \(\frac{1}{ab}\). When dealing with negative exponents, the concept of a reciprocal is crucial. As seen in the step-by-step solution, converting an expression like \(6 r^{-1}\) involves writing it as \(\frac{6}{r}\).

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In the exercise, simplifying algebraic expressions with negative exponents involves understanding the rules for exponents and reciprocals. For instance, transforming \( (6r)^{-1} \) into \(\frac{1}{6r}\). The operations might seem intricate at the beginning, but once the fundamental concepts of exponents and reciprocals are mastered, breaking down algebraic expressions becomes more straightforward.

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91影视

Simplify each expression. (a) \(6 r^{-1}\) (b) \((6 r)^{-1}\) (c) \((-6 r)^{-1}\)

Short Answer

Step by step solution

Simplify part (a)

Simplify part (b)

Simplify part (c)

Key Concepts

Negative Exponents

Reciprocal

Algebraic Expressions

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