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Chapter 5: Problem 48

\((12 y)^{0}\)

Short Answer

Expert verified
(12y)^{0} = 1

Step by step solution

01

Understand the Zero Exponent Rule

According to the zero exponent rule, any non-zero number raised to the power of zero is equal to 1. This can be written mathematically as: \(a^{0} = 1\) where \(a\) is any non-zero real number.
02

Apply the Rule to the Given Problem

In the given problem, we have \( (12 y)^{0} \). Here, \(12y\) is the base that is being raised to the power of zero.
03

Result of the Zero Exponent Rule

Using the zero exponent rule, \( (12 y)^{0} \) simplifies to 1 because any non-zero number raised to the power of zero is 1.

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Key Concepts

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

Zero Exponent Rule
The zero exponent rule is a fundamental principle in algebra. It states that any non-zero number raised to the power of zero equals 1. This can be expressed as: \(a^{0} = 1\) where \(a\) is any non-zero real number. The reasoning behind this rule comes from understanding how exponents work.
For example, consider \(2^3 = 2 \times 2 \times 2 = 8\) and \(2^2 = 2 \times 2 = 4\). If we continue this pattern of decreasing the exponent by 1, we'd have \(2^1 = 2\) and \(2^0 = 1\).
This rule simplifies many algebraic expressions and makes calculations easier.
Exponents
An exponent indicates the number of times a number (the base) is multiplied by itself. For example, \(3^4\) means multiplying 3 by itself 4 times: \(3 \times 3 \times 3 \times 3 = 81\). Exponents are essential in algebra as they help express large numbers compactly and manage repeated multiplications.

Key points about exponents:
  • \(a^m \times a^n = a^{m+n}\) means that when multiplying like bases, you add the exponents.
  • \(a^m / a^n = a^{m-n}\) means that when dividing like bases, you subtract the exponents.
  • \((a^m)^n = a^{m \times n}\) indicates that a power raised to another power multiplies the exponents.

Understanding these properties of exponents helps solve more complex algebraic problems efficiently.
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It includes everything from solving elementary equations to studying abstractions such as groups, rings, and fields. In its most basic form, algebra involves variables, constants, and operators (like addition and multiplication).

Why is algebra important?
  • It provides a universal language to describe mathematical relationships.
  • It allows us to solve equations and find unknown values.
  • It is foundational for advanced studies in mathematics, science, and engineering.

Common tasks in algebra include:
  • Simplifying expressions.
  • Solving for variables in equations.
  • Using formulas to find quantities.
  • Working with polynomials.

By mastering algebra, you build a toolkit for solving an array of practical and theoretical problems.

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Most popular questions from this chapter

\(26,520 \div 85\)

\(\left(2 x^{2}+17 x+21\right) \div(2 x+3)\)

\(2524 \div 4\)

The width of a rectangle is \(10 \mathrm{~cm}\) shorter than the length. a. If \(L=\) length, write a polynomial expression in \(L\) that represents the width, and draw a diagram of the rectangle. Do not include the units. b. Write a polynomial expression in \(L\) that represents the perimeter. c. Write a polynomial expression in \(L\) that represents the area.

\((3 x-7)(3 x+7)\)
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