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Chapter 5: Problem 106

Simplify. Do not use negative exponents in the answer. \(4(-12 y)^{0}\)

Short Answer

Expert verified
The solution is 4.

Step by step solution

01

Apply Exponent Rules

First, we recognize that any expression raised to the power of zero is equal to 1. Therefore, we simplify \((-12y)^{0} = 1\). This changes the expression to \(4 \times 1\).
02

Simplify the Expression

Now that we have simplified \((-12y)^{0}\) to 1, we multiply \(4 \times 1\). This result is simply 4.

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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 concept in algebra that states any non-zero number or expression raised to the power of zero equals one. This might initially seem surprising, but it makes sense when you consider the pattern of exponents. For example:
  • When you have a number with decreasing exponents like: \(a^3 = a \times a \times a\), \(a^2 = a \times a\), \(a^1 = a\), and \(a^0 = 1\), the progression shows how each lower exponent divides by \(a\), always ending in 1 at zero.
This rule applies to any base, whether it鈥檚 a number, a variable, or even a more complex expression. Applying it in our original exercise, \((-12y)^0 = 1\), simplifies the process and allows us to focus on the remaining components of the expression.
Simplifying Expressions
Simplifying expressions involves reducing a complex mathematical expression to its simplest form. This can mean resolving variables, performing operations, or applying mathematical rules, like the Zero Exponent Rule. When we simplify an expression, we're aiming to make calculations easier and the expression clearer.
Returning to our original exercise:
  • The expression was initially given as \(4(-12y)^0\).
  • Thanks to the Zero Exponent Rule, \((-12y)^0\) becomes 1.
  • The expression therefore simplifies to \(4 \times 1\), which is 4.
Simplifying ensures that we understand what is essential in an equation and offers a clearer insight into the properties of the elements involved.
Algebraic Expressions
Algebraic expressions are mathematical phrases containing numbers, variables, and operators. Unlike fixed numbers, variables represent unknown quantities that can change or vary. Algebraic expressions use arithmetic operations such as addition, subtraction, multiplication, and division to combine these elements.
In our exercise, the expression \(4(-12y)^0\) involves:
  • A constant number (4), which is simply a fixed value.
  • A variable expression \((-12y)^0\), which includes a variable (y) that becomes irrelevant when raised to the zero power.
  • The multiplication operator, which connects the constant to the expression evaluated at zero.
This example showcases the versatility of algebraic expressions and their ability to simplify various calculations by adhering to algebraic rules such as exponent rules.

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