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Chapter 6: Problem 504

Simplify. (a) \(\frac{1}{c^{-5}}\) (b) \(\frac{1}{3^{-2}}\)

Short Answer

Expert verified
(a) \(c^{5}\), (b) 9

Step by step solution

01

Understand negative exponents

Negative exponents mean the reciprocal of the base raised to the opposite positive exponent. For example, for any nonzero number 'a' and positive integer 'n', we have: \(a^{-n} = \frac{1}{a^{n}}\).
02

Apply the rule to part (a)

For \(\frac{1}{c^{-5}}\), using the rule of negative exponents, change the exponent from negative to positive: \(c^{-5} = \frac{1}{c^{5}}\) Therefore, \(\frac{1}{c^{-5}} = c^{5}\).
03

Apply the rule to part (b)

For \(\frac{1}{3^{-2}}\), using the rule of negative exponents, change the exponent from negative to positive: \(\frac{1}{3^{-2}} = 3^{2}\).
04

Simplify the expression

Simplify \(3^{2}\): \(3^{2} = 3 \times 3 = 9\).

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Key Concepts

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

Reciprocal
The idea of a reciprocal is fundamental when working with negative exponents. A reciprocal of a number is simply one divided by that number. For instance, the reciprocal of 3 is \(\frac{1}{3}\). Similarly, for any nonzero number 'a', the reciprocal is \(\frac{1}{a}\).

When dealing with exponents, the reciprocal comes into play with negative exponents. For example, \(a^{-n}\) is equivalent to the reciprocal of \(a^n\). This means \(a^{-n}\) can be rewritten as \(\frac{1}{a^n}\). Therefore, negative exponents simplify the expression by converting the base into its reciprocal form raised to a positive exponent.
Positive Exponent
To better understand negative exponents, it鈥檚 essential to grasp what positive exponents are. A positive exponent indicates how many times you multiply the base by itself. For example, if we have \(3^2\), it means 3 multiplied by itself, or \(3 \times 3 = 9\).

Now, when we turn a negative exponent into a positive one, we're essentially finding the reciprocal of the base. The process involves flipping the numerator and the denominator. For instance, \(3^{-2}\) can be rewritten as \(\frac{1}{3^2}\). Changing the exponent from negative to positive simplifies the understanding and calculation of the expression.
Simplify Expression
Simplifying expressions with negative exponents involves a few clear steps. It鈥檚 all about converting the negative exponent into a more workable form. Here's how to simplify effectively:
  • Identify the negative exponent within the expression.
  • Rewrite the expression by taking the reciprocal of the base and changing the exponent to positive.
  • Compute any remaining operations, such as multiplication or division, to simplify the expression fully.
For example, let's take \(\frac{1}{c^{-5}}\). Start by identifying the negative exponent, \(c^{-5}\). Next, rewrite the expression by taking its reciprocal: \(c^{-5} = \frac{1}{c^5}\). Thus, \(\frac{1}{c^{-5}} = c^5\).

Following these easy steps helps to simplify any expressions involving negative exponents, turning them into a form that is easy to understand and solve.

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Most popular questions from this chapter

Divide each polynomial by the binomial. $$\left(y^{2}+7 y+12\right) \div(y+3)$$

Simplify. (a) \(4^{-2}\) (b) \(10^{-3}\)

Divide each polynomial by the binomial. $$\left(x^{2}-14 x+49\right) \div(x-7)$$

Divide each polynomial by the binomial. $$\left(7 q^{3}-5 q-2\right) \div(q-1)$$

Mixed Practice (a) \(\frac{y^{3}}{y}\) (b) \(\frac{y}{y^{3}}\)
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