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Chapter 5: Problem 23

Simplify each numerical expression. \(\left(5^{3}\right)^{-1}\)

Short Answer

Expert verified
The simplified expression is \(\frac{1}{125}\).

Step by step solution

01

Understand the Problem

The expression \(\left(5^{3}\right)^{-1}\) involves finding the negative power of a number. We need to simplify this according to the rules of exponents.
02

Recall the Rule of Negative Exponents

Recall that a negative exponent \(a^{-b}\) means taking the reciprocal of the base raised to the positive power: \(a^{-b} = \frac{1}{a^b}\).
03

Apply the Negative Exponent Rule

Apply the rule to the given expression: \(\left(5^{3}\right)^{-1} = \frac{1}{5^3}\).
04

Calculate the Power

Calculate \(5^3\), which means multiplying 5 by itself three times: \(5 \times 5 \times 5 = 125\).
05

Simplify the Expression

Insert the value of \(5^3\) into the expression in the denominator. Thus, \(\frac{1}{5^3} = \frac{1}{125}\).

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Key Concepts

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

Exponent Rules
When dealing with exponents, it's helpful to remember a set of essential rules that can simplify calculations and make your life easier. Exponents represent repeated multiplication of a base number. That means, for example, that when you see an expression like \(5^3\), it simply means multiply 5 by itself three times: \(5 \times 5 \times 5\).

Here are some key rules to keep in mind:
  • **Multiplication of Powers**: When multiplying two powers with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
  • **Division of Powers**: When dividing two powers with the same base, subtract the exponents: \(a^m \div a^n = a^{m-n}\).
  • **Power of a Power**: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m\times n}\).
  • **Negative Exponents**: A negative exponent \(a^{-b}\) means taking the reciprocal of the base raised to the positive power: \(a^{-b} = \frac{1}{a^b}\).
  • **Zero Exponent**: Any non-zero base raised to the zero power equals one: \(a^0 = 1\).
Understanding these rules can greatly simplify your work with exponents, and explain how a negative exponent translates into a reciprocal in any given expression.
Reciprocal
The concept of a reciprocal is quite simple yet incredibly useful when dealing with exponents, especially negative ones. A reciprocal of a number is basically "flipping" it, or more technically, it's the value that, when multiplied by the original number, results in 1.

For example, the reciprocal of 5 is \(\frac{1}{5}\), because \(5 \times \frac{1}{5} = 1\).
  • **Negative Exponent as a Reciprocal**: A negative exponent indicates you need to take the reciprocal of the base raised to the positive power. So, \(5^{-3}\) becomes \(\frac{1}{5^3}\).
  • The reciprocal approach simplifies expressions, making them easier to understand and solve.
  • Reciprocal operations are common in division and simplifying fractions, linking well with the application of negative exponents.
Whenever you have a negative exponent, think "reciprocal," and it will lead you to the correct simplified form.
Simplifying Expressions
Simplifying expressions is like solving a puzzle where you find the simplest equivalent expression. This involves using rules such as those for exponents and reciprocals to rewrite complex expressions more manageable.

In our example, \((5^3)^{-1}\), you start by applying the negative exponent rule to determine that this means take the reciprocal of \(5^3\):
  • You transform \((5^3)^{-1}\) into \(\frac{1}{5^3}\).
  • Next, calculate the expression \(5^3\) which is \(5 \times 5 \times 5 = 125\).
  • Finally, you simplify the expression to \(\frac{1}{125}\).
This example demonstrates the power of understanding and applying mathematical rules to achieve simplified solutions. Make complex expressions seem more straightforward by approaching them systematically and utilizing the known rules and concepts.

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Most popular questions from this chapter

For Problems \(19-32\), write each of the following in ordinary decimal notation. $$ (5.2)\left(10^{-2}\right) $$

For Problems \(19-32\), write each of the following in ordinary decimal notation. $$ (4.19)\left(10^{3}\right) $$

The field of view of a microscope is \((4)\left(10^{-4}\right)\) meters. If a single cell organism occupies \(\frac{1}{5}\) of the field of view, find the length of the organism in meters. Express the result in scientific notation.

Convert each number to scientific notation and perform the indicated operations. Express the result in ordinary decimal notation. $$ \sqrt{0.00000009} $$

Use your calculator to estimate each of the following. Express final answers in ordinary notation rounded to the nearest one-thousandth. (a) \(1.09^{5}\) (b) \(1.08^{10}\) (c) \(1.14^{7}\) (d) \(1.12^{20}\) (e) \(0.785^{4}\) (f) \(0.492^{5}\)
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