Problem 45 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ a^{0}+2 a^{0}-4 a^{0}, \quad a \neq 0 $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression $$a^0 + 2a^0 - 4a^0$$, given that $$a \neq 0$$. Answer: $$-1$$

Step 1: Apply the property of exponent that a^0 = 1 when a ≠ 0

According to the rules of exponents, any nonzero number raised to the power of 0 equals 1. Since $$a \neq 0$$ in this problem, therefore: $$ a^{0}=1 $$

Step 2: Substitute value of $$a^0$$ in the given expression

Now, substitute the value of $$a^0$$ in the given expression: $$ 1 + 2(1) - 4(1) $$

Step 3: Simplify the expression

Lastly, perform the arithmetic operations: $$ 1+2-4 $$ $$ =3-4 $$ $$ =-1 $$ So, the simplified algebraic expression becomes $$-1$$.

Key Concepts

Exponent Rules

Exponent rules are like the grammar rules of mathematics. They help us determine the result when numbers are involved in repeated multiplication. A crucial rule to remember is that any non-zero number raised to the power of 0 is always 1. This might seem strange at first, but it makes calculations much easier.

For example, in the expression \( a^0 \), as long as \( a eq 0 \), it simplifies to 1. This rule is applicable across different numbers and forms the basis for simplifying expressions with zero exponents.

Understanding this principle allows you to convert any expression with a zero exponent into something more straightforward, as seen in our exercise where \( a^0 = 1 \) was used to simplify the entire expression.

Simplification

Simplification in mathematics involves making an expression as simple as possible, while still keeping its value unchanged. Applying simplification techniques makes solving problems more straightforward.

The first step in the simplification process is replacing each part of the expression with an equivalent but simpler version. By doing this, we make complex expressions easier to handle. For instance, transitioning from \( a^0 \) to 1 helps in reducing extra complexity.

Once this simplification is performed, you can focus on solving the basic arithmetic involved. A useful tip: always simplify parts of the expression first before dealing with calculation to avoid errors.

Arithmetic Operations

Arithmetic operations are the backbone of math, consisting mainly of addition, subtraction, multiplication, and division. These operations are used to solve mathematical expressions.

In the example exercise, after replacing \( a^0 \) with 1, we had a simpler expression: \( 1 + 2(1) - 4(1) \). Each term needs to be calculated to arrive at the simplest form. Performing the operations sequentially is essential:

• First, resolve the multiplications, turning the expression into \( 1 + 2 - 4 \).
• Next, handle additions and subtractions \( 1 + 2 \) and then \( 3 - 4 \).
• The final output is \(-1\).

Ensuring that arithmetic operations are applied correctly ensures the accuracy of any simplified expression. Stick to the basic order and proceed step by step for clarity and correctness.