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Chapter 5: Problem 24

Assume that all variables represent nonzero real numbers. $$ 19^{0} $$

Short Answer

Expert verified
19^0 = 1

Step by step solution

01

- Identify the Base and Exponent

The given expression is 19^{0}. Here, the base is 19 and the exponent is 0.
02

- Apply the Zero Exponent Rule

According to the zero exponent rule, any nonzero real number raised to the power of 0 is equal to 1. Mathematically, for any nonzero real number a, a^0 = 1.
03

- Simplify Using the Rule

Using the rule from step 2, simplify 19^0 to get the final answer. Therefore, 19^0 = 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 way to express repeated multiplication of a number by itself. For example, in the expression \(3^4\), the base is 3 and the exponent is 4. This means you multiply 3 by itself 4 times: \(3^4 = 3 \times 3 \times 3 \times 3 = 81\). Since exponents simplify the way we write large numbers, they are very handy in mathematics.
Understanding exponents is crucial because they serve as the building blocks for more complex algebraic concepts.
  • Basic exponent rules are essential for simplifying expressions and solving equations efficiently.
  • They help in various areas of math, such as algebra, calculus, and even in real-life applications like computing and finance.
Base and Exponent
In any expression involving exponents, it is important to identify the base and the exponent properly.
  • The base is the number that is being multiplied.
  • The exponent indicates how many times the base is multiplied by itself.
In the expression \(19^0\), 19 is the base, and 0 is the exponent.
Here, understanding the role of both the base and exponent helps in applying the appropriate mathematical rules. For example, there's a useful rule known as the Zero Exponent Rule. This rule states: For any nonzero number \(a\), \(a^0 = 1\). So in our example, \(19^0\) simplifies to 1.
Mastering these core concepts will make it easier to solve more complex algebraic problems involving exponents.
Simplification of Expressions
Simplifying algebraic expressions is a fundamental skill in math. It involves reducing the expression to its simplest form while preserving its value. When dealing with exponents, certain rules and properties apply.
One key rule is the Zero Exponent Rule, which states that any nonzero number raised to the power of 0 is 1. For example, in the given problem \(19^0 = 1\), we use this rule to simplify the expression.
Understanding simplification allows us to solve equations faster and with fewer errors. Here are some steps to simplify expressions:
  • Identify the base and the exponent.
  • Apply exponent rules appropriately.
  • Ensure the expression is in its simplest form.
This approach ensures clarity and accuracy in mathematical calculations.

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Most popular questions from this chapter

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ -4 r^{-2}\left(r^{4}\right)^{2} $$

Two expressions are given. Replace x with 3 and y with 4 to show that, in general, the two expressions are not equivalent. $$ (x+y)^{2} ; \quad x^{2}+y^{2} $$

Evaluate. Express answers in standard notation. $$ \frac{3 \times 10^{-2}}{12 \times 10^{3}} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \frac{\left(p^{-2}\right)^{0}}{5 p^{-4}} $$

Solve each problem. The speed of light is approximately \(3 \times 10^{10} \mathrm{cm}\) per sec. How long will it take light to travel \(9 \times 10^{12} \mathrm{cm} ?\)
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