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Chapter 10: Problem 19

Simplify. $$a^{0}$$

Short Answer

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Step by step solution

01

Understanding the Zero Exponent Rule

The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. This can be written as: \(\text{For any } a eq 0, \text{ we have } a^0 = 1\)
02

Apply the Rule

Applying the zero exponent rule to the given expression, we get:\[a^{0} = 1\]

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Key Concepts

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

Exponents
Exponents are a fundamental part of algebra and mathematics in general. They represent repeated multiplication of a base number. For example, in the expression \(2^3\), the base is 2 and the exponent is 3, meaning you multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\). Exponents can be positive, negative, or even zero. Understanding how to work with exponents is key to handling more complex algebra problems. Let's look at the zero exponent rule specifically, which is very important in simplifying expressions:
  • For any non-zero number \(a\), the expression \(a^0\) always equals 1. This is true for any base, whether it's a variable like \(a\) or a number like 5.
This rule helps simplify many mathematical expressions easily.
Simplification
Simplification is the process of making an algebraic expression easier to understand or solve. By applying rules, such as the zero exponent rule, you can simplify complex expressions into more manageable ones. For example, with the expression \(a^0\), the zero exponent rule tells us it simplifies to 1. Here's how simplification works in different cases:
  • Combining like terms: Expressions like \(3x + 2x\) can be simplified to \(5x\).
  • Applying exponent rules: Expressions like \((x^2)^3\) simplify to \(x^6\) using the power rule for exponents.
  • Canceling terms: In fractions, terms in the numerator and denominator can cancel each other out to simplify the fraction.
Simplification is a powerful tool for making problems manageable, ensuring that your solutions are accurate and easier to understand.
Algebra Basics
Getting a strong grasp of algebra basics is crucial for solving more advanced math problems. In algebra, we often work with variables, constants, and various operations like addition, subtraction, multiplication, and division. Here are some core principles to remember:
  • Variables are symbols that represent unknown values. They give us the ability to write general equations and formulas.
  • Operations follow specific rules, such as the order of operations (PEMDAS), which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
Understanding these basics allows you to effectively manipulate and solve equations. With algebra, the goal is often to isolate the variable (or unknown) to find its value, making use of techniques such as simplification and applying specific math rules like the zero exponent rule.

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Most popular questions from this chapter

Write the opposite of the polynomial. $$12 x^{3}+4 x^{2}-8$$

Which is greater? \(\left(2^{3}\right)^{2}\) or \(\left(2^{2}\right)^{3}\)

Simplify. Write the answers with positive exponents only. $$\left(p^{-4}\right)^{-5}$$

Simplify. Write the answer with positive exponents only. $$\frac{2}{x^{-2} y^{-3}}$$

Simplify. Write the answers with positive exponents only. $$\left(-3 c^{2} d^{-3}\right)\left(6 c^{3} d^{-4}\right)$$
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