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Chapter 4: Problem 10

Decide whether each expression is equal to \(0,1,\) or \(-1 .\) $$ (-11)^{0}-11^{0} $$

Short Answer

Expert verified
The expression \((-11)^{0} - 11^{0}\) equals 0.

Step by step solution

01

Identify the Exponents

Notice that in the expressions \((-11)^{0}\) and \(11^{0}\), the exponents are both 0.
02

Apply the Zero Exponent Rule

Use the property that any non-zero number raised to the power of 0 is 1. Therefore, \((-11)^{0} = 1\) and \(11^{0} = 1\).
03

Subtract the Results

Now subtract the two results: \(1 - 1 = 0\).
04

Conclusion

The value of the expression \((-11)^{0} - 11^{0}\) is 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 fundamental concept in mathematics, particularly in algebra. They are used to represent repeated multiplication of a number by itself. For example, the expression \(2^3\) means \(2\) multiplied by itself \(3\) times: \(2 \times 2 \times 2 = 8\). This short form of writing repeated multiplication makes it much easier to work with large numbers.
An exponent is typically written as a small number (called the power) to the upper right of the base number. Here's a breakdown:
  • In the expression \(3^4\), \(3\) is the base, and \(4\) is the exponent.
  • This means \(3\) is multiplied by itself 4 times: \(3 \times 3 \times 3 \times 3 = 81\).
Exponents are not limited to positive numbers; they can also be negative or zero, which leads us to the next important concept: the zero exponent property.
Zero Exponent Property
The zero exponent property states that any non-zero number raised to the power of zero is equal to one.
This may seem surprising at first, but it is a crucial rule in algebra. Here's why this works:
  • Consider the sequence of powers of 2: \(2^3 = 8\), \(2^2 = 4\), \(2^1 = 2\).
  • Notice that each time the exponent decreases by 1, the result is divided by 2.
  • Following this pattern, \(2^0\) would be \(2 \div 2 = 1\).
The rule works for any non-zero number: \(a^0 = 1\).
It's important to note that \(0^0\) is usually considered indeterminate in mathematics. For your exercise, applying the zero exponent property to \((-11)^0\) and \(11^0\) gives us:
  • \((-11)^0 = 1\)
  • \(11^0 = 1\)
Then, subtracting these results gives us the final answer of 0.
Basic Algebra
Algebra is a branch of mathematics that uses letters and symbols to represent numbers and quantities in formulas and equations. Basic algebra involves:
  • Understanding variables: Symbols (like \(x\) or \(y\)) used to represent unknown values
  • Operations: Addition, subtraction, multiplication, and division involving these variables
  • Properties and rules, such as the distributive property and exponent rules
The zero exponent property is one of many rules we use in algebra to simplify expressions and solve equations.
In the given exercise, you used basic algebra to simplify the expression \((-11)^0 - 11^0\) by applying the zero exponent property. You identified the exponents, applied the rule, and then performed the subtraction. This step-by-step method is a key part of understanding and solving algebraic problems, making complex expressions much more manageable.

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Most popular questions from this chapter

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