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Chapter 10: Problem 233

In the following exercises, simplify. $$a^{0}$$

Short Answer

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Step by step solution

01

Identify the Base and Exponent

Identify the base and exponent in the given expression. Here, the base is 'a' and the exponent is 0.
02

Understand the Zero Exponent Rule

Recall the zero exponent rule, which states that any nonzero number raised to the power of 0 is equal to 1. Mathematically, this is expressed as follows: a^{0} = 1.
03

Apply the Zero Exponent Rule

Apply the rule to the given expression. Since the base 'a' (assuming 'a' is a nonzero number) is raised to the 0 power, the expression simplifies to 1.

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Key Concepts

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

Base and Exponent
Before diving into any algebraic expressions, it's crucial to understand the concepts of base and exponent. The 'base' in an exponentiation refers to the number that is being multiplied. The 'exponent', sometimes called the power, tells you how many times to multiply the base by itself. For example, in the expression \(3^4\), the base is 3, and the exponent is 4. This means we multiply 3 by itself 4 times to get 3 * 3 * 3 * 3 = 81.
In the initial exercise, the base is 'a', and the exponent is 0. Recognizing these parts helps you understand what the expression represents and how to simplify it.
Simplifying Expressions
Simplifying expressions is a fundamental skill in algebra. It involves reducing an expression to its simplest form. This can make it easier to work with and understand.
For example, if you have a complex equation, breaking it down into simpler parts can make it more manageable. Often, rules like the Zero Exponent Rule come into play to help us simplify expressions. By applying known rules and properties, we can transform a complicated expression into something that's much simpler.
In the given exercise, applying the zero exponent rule transforms \(a^0\) into 1, which is much simpler and easier to understand.
Power of Zero in Algebra
The Power of Zero is a unique and critical concept in algebra. The Zero Exponent Rule states that any nonzero number raised to the power of zero is always 1. Mathematically, this is expressed as \(a^0 = 1\), where 'a' is any nonzero number. This can sometimes seem counterintuitive, but it is a fundamental principle in mathematics.
Imagine you are repeatedly multiplying a number by itself. If you do this zero times, you're essentially left with the multiplicative identity, which is 1.
  • This principle applies universally to all nonzero bases.
  • It's a handy rule for simplifying expressions quickly.
By applying this rule, you can turn the expression \(a^0\) into 1 without needing to perform any actual multiplication, simplifying the problem considerably.

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Most popular questions from this chapter

Simplify each expression. $$\frac{8 y^{2}-16 y+20}{4 y}$$

Multiply the binomials using various methods. $$(5 u-3)(u+8)$$

Divide the monomials. $$\frac{-30 x^{8}}{-36 x^{9}}$$

Convert each number to decimal form. $$1.5 \times 10^{8}$$

Simplify. $$\frac{u^{3}}{u^{2} \cdot u^{4}}$$
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