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Chapter 4: Problem 14

Decide whether each expression is equal to \(0,1,\) or \(-1 .\) $$ 6^{0}-13^{0} $$

Short Answer

Expert verified
The expression evaluates to 0.

Step by step solution

01

Evaluate Exponents

First, determine the value of each exponent in the expression. Recall the property that any non-zero number raised to the power of 0 is 1. Hence, examine the terms separately: \( 6^0 = 1 \) and \( 13^0 = 1 \).
02

Calculate the Difference

Subtract the results of the evaluated exponents from each other. That is, perform the operation \( 1 - 1 \).
03

Simplify the Expression

Simplify the result of the subtraction: \( 1 - 1 = 0 \). Therefore, the value of the expression \( 6^0 - 13^0 \) is equal to 0.

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Key Concepts

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

Exponentiation Rules
In mathematics, exponentiation is a popular operation involving numbers called bases and exponents. The base is the number that gets multiplied, while the exponent tells how many times the base is used as a factor. For example, in the expression \( 2^3 \), 2 is the base, and 3 is the exponent, meaning \( 2 \times 2 \times 2 = 8 \).
When evaluating expressions involving exponents, a key rule to remember is that any non-zero number raised to the power of 0 is always 1. This helps simplify problems significantly. So, for any base \(a\), \( a^0 = 1 \). For example, \( 6^0 = 1 \) and \( 13^0 = 1 \).
Understanding these rules helps you handle more complex expressions efficiently!
Simplifying Expressions
Simplifying an expression means reducing it to its simplest form. This is often necessary in algebra to find the most straightforward form of an equation or expression. Suppose we have evaluated an expression like \( 6^0 - 13^0 \) by following exponentiation rules. Both exponents simplify to 1, leaving us with \( 1 - 1 \).
Next, you perform arithmetic operations such as subtraction, which further simplifies the equation. These basic steps are crucial for dealing with algebraic expressions in both basic and advanced mathematics contexts.
Subtraction
Subtraction is one of the four basic arithmetic operations and signifies taking one number away from another. In algebra, subtraction is used to simplify expressions, solve equations, and more. Take the example from our exercise: after applying the exponentiation rules, we are left with the subtraction \( 1 - 1 \). Performing this operation results in 0.
Basic subtraction rules ensure that you consistently achieve accurate results in your expressions. Understanding these rules is crucial, especially when subtracting similar terms or constants.
By mastering subtraction, you can handle all kinds of mathematical problems with ease!

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Most popular questions from this chapter

Our system of numeration is called a decimal system. In a whole number such as 2846 each digit is understood to represent the number of powers of 10 for its place value. The 2 represents two thousands \(\left(2 \times 10^{3}\right),\) the 8 represents eight hundreds \(\left(8 \times 10^{2}\right),\) the 4 represents four tens \(\left(4 \times 10^{1}\right),\) and the 6 represents six ones (or units) \(\left(6 \times 10^{\circ}\right)\) \(2846=\left(2 \times 10^{3}\right)+\left(8 \times 10^{2}\right)+\left(4 \times 10^{1}\right)+\left(6 \times 10^{0}\right) \quad\) Expanded form $$ \text {Keeping this information in mind,} $$ Divide the polynomial \(2 x^{3}+8 x^{2}+4 x+6\) by 2

Evaluate. $$ 100(72.79) $$

Subtract. $$ \begin{array}{r} {x^{5}+x^{3}-2 x^{2}+3} \\ {-4 x^{5}+3 x^{2}-8} \\ \hline \end{array} $$

Simplify by writing each expression wth positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{\left(2 y^{-1} z^{2}\right)^{2}\left(3 y^{-2} z^{-3}\right)^{3}}{\left(y^{3} z^{2}\right)^{-1}} $$

A polynomial in the variable \(x\) has degree 6 and is divided by a monomial in the variable \(x\) having degree \(4 .\) What is the degree of the quotient?
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