Problem 68 - Algebra A Combined Function

Question

Simplify each expression. $$ -2 x^{0} $$

Step-by-Step Solution

Answer

The simplified expression is \(-2\).

Step 1: Recall the Power of Zero Rule

Any number or variable raised to the power of zero is equal to 1. Thus, the expression \(x^0\) simplifies to 1.

Step 2: Multiply Coefficients

Substitute the simplified value from Step 1 into the original expression. This turns \(-2x^0\) into \(-2 \times 1\).

Step 3: Simplify the Expression

Perform the multiplication: \(-2 \times 1 = -2\). This gives us the final simplified expression.

Key Concepts

Power of Zero Rule

The Power of Zero Rule is a fundamental concept in mathematics that greatly simplifies expressions involving exponents. According to this rule, any non-zero number raised to the power of zero equals one. This might seem surprising at first, but it is consistent with the pattern of exponents. For example, consider how powers of a number work: \(x^3 = x \times x \times x\), \(x^2 = x \times x\), and \(x^1 = x\). As you decrease the exponent by one, you remove a factor of \(x\), and it naturally follows that when you reduce the exponent to zero, you "remove" all factors of \(x\), resulting in \(x^0 = 1\).

• This rule applies to all non-zero numbers and variables.
• It simplifies expressions and is a useful tool in algebra.

When working with expressions like \(-2x^0\), you can apply this rule directly to simplify \(x^0\) to 1, hence making the problem easier to solve.

Multiplying Coefficients

Once you have simplified any terms within an expression, particularly those involving exponents, the next step involves multiplying coefficients. In algebra, coefficients are the numerical parts of terms. When a variable part of a term is simplified, such as through the Power of Zero Rule, you focus on the coefficient.

• In the expression \(-2x^0\), the coefficient is \(-2\).
• After simplifying \(x^0\) to \(1\), the expression becomes \(-2 \times 1\).

Multiplying coefficients is straightforward: simply multiply the numbers together. In this case, multiplying \(-2\) by \(1\) results in \(-2\). The process is important for maintaining the numerical relationship in algebraic expressions.

Exponents and Powers

Exponents and powers are mathematical tools used to describe repeated multiplication. An exponent tells you how many times to multiply the base by itself. For instance, in the term \(x^3\), \(x\) is the base and \(3\) is the exponent, meaning that \(x\) is multiplied by itself three times: \(x \times x \times x\). Recognizing and applying the rules that govern exponents is crucial for simplifying expressions.

• An important rule is that any non-zero number to the zero power is one, as discussed in the Power of Zero Rule.
• Understanding how to handle coefficients separately from variables aids in properly simplifying and solving expressions.
• Mastering these concepts is fundamental to progressing in algebra and higher-level math.

By knowing how to manage exponents and powers, you can simplify complex expressions and make calculations more manageable. In the given exercise, applying the exponent rules allowed us to simplify the expression efficiently.