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Chapter 7: Problem 5

Exercises \(1-6\) consist of a number of related parts. Simplify each as completely as possible. (a) \(3-4\) (b) \(x^{3} x^{-4}\) (c) \(3(-4)\) (d) \(\left(x^{3}\right)^{-4}\) (e) \(3-(-4)\) (f) \(\frac{x^{3}}{x^{-4}}\) (g) \(3^{-4}\)

Short Answer

Expert verified
-1, \(\frac{1}{x}\), -12, \(\frac{1}{x^{12}}\), 7, \(x^7\), \(\frac{1}{81}\)

Step by step solution

01

Simplify (a)

For part (a), calculate the result of the expression by performing simple subtraction. The expression is: \[ 3 - 4 \ = -1 \]
02

Simplify (b)

For part (b), simplify the expression by using the properties of exponents. The expression is: \[ x^{3} x^{-4} \ = x^{3 + (-4)} \ = x^{-1} \ = \frac{1}{x} \]
03

Simplify (c)

For part (c), calculate the result of the multiplication. The expression is: \[ 3(-4) \ = -12 \]
04

Simplify (d)

For part (d), simplify the expression by using the properties of exponents. The expression is: \[ \big(x^3\big)^{-4} \ = x^{3 \times (-4)} \ = x^{-12} \ = \frac{1}{x^{12}} \]
05

Simplify (e)

For part (e), calculate the result of the expression by performing addition since subtracting a negative number is equivalent to adding the positive number. The expression is: \[ 3 - (-4) \ = 3 + 4 \ = 7 \]
06

Simplify (f)

For part (f), simplify the expression by using the properties of exponents. The expression is: \[ \frac{x^3}{x^{-4}} \ = x^{3 - (-4)} \ = x^{3 + 4} \ = x^7 \]
07

Simplify (g)

For part (g), use the fact that any number raised to the power of a negative exponent is the reciprocal of the number raised to the positive exponent. The expression is: \[ 3^{-4} \ = \frac{1}{3^4} \ = \frac{1}{81} \]

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Key Concepts

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

simplification of expressions
Simplifying expressions is a fundamental skill in elementary algebra. It often involves combining like terms, using properties of numbers or variables, and performing basic arithmetic operations.
Let's walk through some examples from the exercise:
  • For part (a), '3-4', we see a simple subtraction. Simplifying this, we get: 3 - 4 = -1.

  • For part (e), '3-(-4)', we need to remember that subtracting a negative number is the same as adding its positive counterpart. This results in: 3 - (-4) = 3 + 4 = 7.
The key here is understanding and applying basic arithmetic principles to simplify equations.
properties of exponents
Learning the properties of exponents is crucial for handling expressions involving powers of numbers or variables. Here are some properties that were applied in the exercise:
  • Product of Powers: When multiplying two exponents with the same base, add their exponents: \(x^a \times x^b = x^{a+b}\). For example, in part (b) 'x^3 x^{-4}', the exponents are added: x^{3 + (-4)} = x^{-1} = \(\frac{1}{x}\).

  • Power of a Power: When raising an exponentiated base to another power, multiply the exponents: \(\big(x^a\big)^b = x^{a \times b}\). In part (d), \(\big(x^3\big)^{-4}\) is simplified to x^{3 \times (-4)} = x^{-12} = \(\frac{1}{x^{12}}\).

  • Quotient of Powers: When dividing two exponents with the same base, subtract their exponents: \(\frac{x^a}{x^b} = x^{a - b}\). In part (f), \(\frac{x^{3}}{x^{-4}}\) simplifies to x^{3 - (-4)} = x^{7}.

  • Negative Exponents: A negative exponent indicates a reciprocal: \(a^{-n} = \frac{1}{a^n}\). In part (g), 3^{-4} becomes \(\frac{1}{3^4}\) = \(\frac{1}{81}\).
Understanding these properties helps in breaking down and simplifying complex expressions efficiently.
basic arithmetic operations
Basic arithmetic operations like addition, subtraction, multiplication, and division form the foundation of algebra.
Let's consider:
  • Subtraction: In part (a), the operation '3-4' simplifies directly to -1. For part (e), '3-(-4)', subtracting a negative number converts into addition, making it 3 + 4 = 7.

  • Multiplication: In part (c), '3(-4)' illustrates multiplication of a positive by a negative number, resulting in -12.

  • Division: Although not directly in the operations, understanding how to handle reciprocal expressions like \(\frac{x^3}{x^{-4}}\) in part (f) involves underlying division principles.
Mastering these basic operations is essential as they are woven into more complex algebraic manipulations.

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Most popular questions from this chapter

Maria can make five flower arrangements per hour, while Francis can make seven flower arrangements per hour. If Maria starts working at 9: 00 A.M. and is joined by Francis at 11: 00 A.M., at what time will they have made 70 flower arrangements?

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