Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression for the indicated value of \(x\).\(8 x^{0}-(8 x)^{0} \quad x=-7\)

Step-by-Step Solution

Answer

The evaluated expression is 7

Step 1: Substitution and Simplification - Step 1

First, let's substitute \(x = -7\) into the expression. This gives us: \(8(-7)^{0} - (8(-7))^{0}\)

Step 2: Applying Exponent Property - Step 2

Next, recall the property: any number (except 0) raised to the power of 0 is 1. So, \((-7)^{0} = 1\) and \((8(-7))^{0} = 1\). Substituting these back into our expression, we get: \(8 * 1 - 1 = 8 - 1 = 7\)

Key Concepts

Exponent Properties

When dealing with exponents, there are a few key properties that make simplifying expressions easier. One important rule is that any non-zero number raised to the power of zero is always equal to 1. This might seem confusing at first, but it's a fundamental rule in mathematics.
For example, no matter what number you have, say 5, raising it to the power of 0 will give you 1, so:

• \( 5^0 = 1 \)
• \( (-7)^0 = 1 \)

This also holds true even if the expression inside the parenthesis is negative or a complex expression. As mentioned in our problem, both \((-7)^0\) and \((8(-7))^0\) equal 1. Understanding this rule helps tremendously when simplifying expressions as it often reduces computation to basic arithmetic.

Expression Substitution

Expression substitution is an essential technique in algebra that involves replacing variables with known values to simplify the expression. This method directly affects the calculation results if performed correctly.
For instance, in our problem, the variable \(x\) was given with a particular value of \(-7\). To find the value of the entire expression, we substitute \(x\) with \(-7\). This changes our expression from:

• \( 8x^0 - (8x)^0 \)

to

• \( 8(-7)^0 - (8(-7))^0 \)

Substituting ensures that we are evaluating the expression accurately based on the value given.

Algebraic Simplification

Simplifying algebraic expressions makes it easier to work with them and understand their values. After substituting and understanding exponent rules, simplifying becomes the next logical step.
In the problem, once we substituted \(x = -7\) and applied the exponent properties, the expression simplified to basic arithmetic:

• \( 8 \times 1 - 1 \)

Here's a step-by-step breakdown:

• Any expression \(x^0\) turned into 1 due to our exponent rule.
• Multiplication comes next, where \(8 \times 1 = 8\).
• Finally, subtract the numbers: \(8 - 1 = 7\).

Simplification reduces complexity, revealing that the expression evaluates to 7. This process highlights the power of breaking down expressions into simpler component parts.