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Chapter 5: Problem 76

Simplify each expression. See Example \(10 .\) $$ -3^{0}+4^{0} $$

Short Answer

Expert verified
The simplified expression is 0.

Step by step solution

01

Recall the Zero Exponent Rule

According to the zero exponent rule, any non-zero base raised to the exponent of zero equals one. Therefore, for any number \(a\), \(a^0 = 1\).
02

Apply the Zero Exponent Rule

Apply the zero exponent rule to each term in the expression:\(-3^0 + 4^0\). Since \(3^0 = 1\) and \(4^0 = 1\), the expression becomes:\(-1 + 1\).
03

Simplify the Expression

Simplify the resulting expression by performing the addition:\(-1 + 1 = 0\). Therefore, the simplified form of the expression is 0.

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is a fundamental skill in algebra that involves reducing expressions to their simplest form. When simplifying, you aim to make expressions easier to understand and work with. This can involve several steps, such as applying mathematical rules and combining like terms.

In the given exercise, the expression \( -3^0 + 4^0 \) needs to be simplified. By understanding and applying algebraic rules, we can reduce this to a simpler form. The process often involves:
  • Finding common factors or terms
  • Applying exponent rules, such as the zero exponent rule
  • Performing basic arithmetic operations
Once simplified, expressions provide a clearer and more manageable form, highlighting their core value or meaning.
Exponentiation
Exponentiation is a mathematical operation involving two numbers, the base and the exponent. It signifies the base multiplied by itself a specific number of times, indicated by the exponent.

Consider the term \(-3^0\). Normally, the base \(-3\) would be used to multiply itself the number of times equal to the exponent. However, due to the zero exponent rule, when any non-zero number is raised to the power of zero, the result is always 1. This seems counterintuitive at first, but a clear understanding of exponentiation helps in grasping such rules.

Knowing how exponents work helps in simplifying complex expressions efficiently, making it a crucial element in algebra.
Algebraic Rules
Algebraic rules, like the zero exponent rule, are key principles that guide the simplification and manipulation of algebraic expressions. These rules are like mathematical shortcuts that help to break down complex expressions.

In algebra, you may come across several essential rules:
  • Zero Exponent Rule: Any non-zero number raised to zero is 1, \( a^0 = 1 \).
  • Product of Powers Rule: For bases that are the same, add their exponents, \( a^m \times a^n = a^{m+n} \).
  • Quotient of Powers Rule: For bases that are the same, subtract their exponents, \( \frac{a^m}{a^n} = a^{m-n} \).
  • Power of a Power Rule: Multiply the exponents, \( (a^m)^n = a^{m \cdot n} \).
Each of these rules has unique importance and, when appropriately applied, can greatly simplify the process of working with expressions. For example, in our original problem, recognizing and using the zero exponent rule allowed us to streamline the expression down to its simplest form, 0.

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Most popular questions from this chapter

Simplify the following. See Section \(5.1 .\) $$ \frac{27 y^{14}}{3 y^{7}} $$

The perimeter of a square is \(\left(12 x^{3}+4 x-16\right)\) feet. Find the length of its side.

A piece of quarter-round molding is \((13 x-7)\) inches long. If a piece \((2 x+2)\) inches is removed, express the length of the remaining piece of molding as a polynomial in \(x .\) (GRAPH NOT COPY)

Simplify each expression. See Section \(5.1 .\) $$ \frac{-48 a b^{6}}{32 a b^{3}} $$

Divide If the divisor contains 2 or more terms, use long division. $$ \frac{y^{3}+3 y^{2}+4}{y-2} $$
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