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Chapter 4: Problem 37

Evaluate. Assume that all variables represent nonzero real numbers. $$ -3^{0}+3^{0} $$

Short Answer

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Step by step solution

01

- Evaluate each exponent

Calculate the value of each exponent separately. Recall that any nonzero number raised to the power of 0 is 1. So, evaluate each term:\[3^{0} = 1\]
02

- Apply the exponent rule

Since we have two identical exponents, compute each term separately:\[-3^{0} = -1\]\[3^{0} = 1\]
03

- Combine the terms

Now, add the two results from the previous step together:\[-1 + 1 = 0\]

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Key Concepts

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

Understanding Exponents
Exponents are a way to represent repeated multiplication of the same number. For example, instead of writing 3 * 3 * 3, you can write it as 3 raised to the power of 3, or 33. The number that gets multiplied is called the base, and the number of times it鈥檚 multiplied is called the exponent. When working with exponents, it's essential to understand some basic rules, such as:
  • Product Rule: \(a^m \times a^n = a^{m+n}\)
  • Power Rule: \((a^m)^n = a^{mn}\)
  • Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\) if a 鈮� 0
These rules help simplify expressions and solve problems involving exponents more efficiently.
Working with Algebraic Expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. For instance, 2x + 5 is an algebraic expression where 2 is the coefficient, x is the variable, and 5 is the constant. To work with algebraic expressions effectively:
  • Simplify expressions by combining like terms
  • Apply the distributive property \(a(b+c) = ab+ac\)
  • Understand the significance of each component
Understanding these basics can make evaluating expressions with exponents, like the given problem, much easier.
The Zero Exponent Rule
The zero exponent rule states that any nonzero number raised to the power of zero is equal to one: \(a^0 = 1\), where a 鈮� 0. This rule simplifies many expressions and comes in very handy when solving problems. For example, in the expression \(-3^0 + 3^0\), you apply this rule to both terms:
  • \(-3^0 = -1\)
  • \(3^0 = 1\)
Combining these, you get \(-1 + 1 = 0\). Hence, remembering that anything to the zero power equals one is a useful trick in algebra.

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Most popular questions from this chapter

Find each product. $$ 5 y(-6 y+1) $$

Solve each problem. In 2017 , the population of Costa Rica was approximately \(4.93 \times 10^{6} .\) The population density was 96.6 people per square kilometer. (Data from The World Almanac and Book of Facts.) (a) Write the population density in scientific notation. (b) To the nearest square kilometer, what is the area of Costa Rica?

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(5^{-3}\right)^{-2} $$

Write each number in standard notation. $$ 5.92 \times 10^{0} $$

Apply the quotient rule for exponents, if possible, and write each result using only positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{5^{-4}}{5^{2}} $$
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