Problem 46 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 4 x^{0}+3 x^{0}-5 x^{0}+7 x^{0}-x^{0}, \quad x \neq 0 $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$4x^0+3x^0-5x^0+7x^0-x^0$$ when x ≠ 0. Answer: The simplified expression is 8, when x ≠ 0.

Step 1: Substitute the value of x raised to the power 0

First, remember that any non-zero number raised to the power of 0 is equal to 1. So, x^0 = 1 when x ≠ 0. Let's substitute this value into the given expression: $$ 4x^0 + 3x^0 - 5x^0 + 7x^0 - x^0 = 4(1) + 3(1) - 5(1) + 7(1) - (1) $$

Step 2: Perform the arithmetic operations

Now that we have substituted the value of x^0, we just need to perform the addition and subtraction of these numbers: $$ 4(1) + 3(1) - 5(1) + 7(1) - (1) = 4 + 3 - 5 + 7 - 1 $$

Step 3: Calculate the final simplified expression

Perform the addition and subtraction from left to right: $$ (4 + 3) - 5 + 7 - 1 = 7 - 5 + 7 - 1 $$ $$ (7-5) + 7 - 1 = 2 + 7 - 1 $$ $$ (2 + 7) - 1 = 9 - 1 $$ $$ 9 - 1 = 8 $$ So after simplifying the given algebraic expression, we end up with the final answer: $$ 4x^0+3x^0-5x^0+7x^0-x^0 = 8, \quad x \neq 0 $$

Key Concepts

Simplification

Simplification is a fundamental concept in algebra that involves reducing expressions to their simplest form. This process makes complex equations more manageable and easier to work with.
In the context of the exercise, simplification involved replacing each term raised to the power of zero with the number one, because any non-zero number raised to the power zero is always one.
Once these substitutions were made, the expression included only integers to allow for straightforward arithmetic operations.

Exponents

Understanding exponents is crucial in simplifying algebraic expressions. An exponent indicates how many times a number, known as the base, is multiplied by itself.
Here are some key points:

• When a base is raised to the power of zero, the result is always one, given the base is not zero.
• This is known as the zero-exponent rule.
• For example, in expressions like \(x^0\), where \(xeq0\), we can confidently replace \(x^0\) with 1.

This property allows algebraic expressions to be simplified by substituting exponents with integers, significantly reducing their complexity.

Arithmetic Operations

Arithmetic operations form the bedrock for solving and simplifying algebraic expressions. They include addition, subtraction, multiplication, and division.
In our exercise, once the substitution for the exponent was completed, the focus shifted to simple arithmetic:

• First, perform additions and subtractions as per the sequence defined by the expression.
• Keep in mind the importance of operational hierarchy or order of operations (also known as PEMDAS) - even in simple additions and subtractions.

Performing these operations correctly is essential since failing to follow the correct order can lead to incorrect answers.