Problem 17 - Elementary and Intermediate Algebra

Question

Simplify each expression. \(2 x^{0}\)

Step-by-Step Solution

Answer

The expression simplifies to 2.

Step 1: Understand the Power of Zero Rule

The power of zero rule states that any non-zero number raised to the power of zero is equal to 1. This means that no matter what the base is, if the exponent is 0, the value simplifies to 1.

Step 2: Apply the Power of Zero Rule

Identify the part of the expression that has an exponent. Here, it is \(x^{0}\). Since \(x\) is raised to the power of 0, according to the rule, \(x^{0} = 1\).

Step 3: Substitute and Simplify

Replace \(x^{0}\) with 1 in the expression. Hence, the expression becomes \(2 \cdot 1\).

Step 4: Perform the Multiplication

Now, multiply the coefficient 2 by the simplified value 1: \(2 \times 1 = 2\). This is the simplified form of the original expression.

Key Concepts

Power of Zero

When dealing with algebraic expressions, a special rule arises called the **Power of Zero**. This rule states that any non-zero number raised to the power of zero is equal to 1. This might initially be counterintuitive, but it helps simplify expressions significantly. For instance, if you have an expression like \(2x^0\), the part of the expression \(x^0\) equates to 1 regardless of what \(x\) is (as long as \(x\) is not zero). Thus, you replace \(x^0\) with 1, turning the entire expression into a simpler form.

Exponent Rules

Exponent rules are fundamental to simplifying expressions and are easy to learn once you understand the basics. The **Power of Zero** rule is just one of the rules governing exponents. Other key rules include:

• **Product of Powers Rule**: When multiplying numbers with the same base, add their exponents: \(a^m imes a^n = a^{m+n}\).
• **Power of a Power Rule**: When you raise an exponent to another power, multiply the exponents: \((a^m)^n = a^{m imes n}\).
• **Power of a Product Rule**: When raising a product to a power, distribute the exponent to each part of the product: \((ab)^n = a^n imes b^n\).

These rules help you streamline calculations and simplify expressions much more easily.
When you encounter exponents in any mathematical problem, applying these rules consistently will guide you towards the simplest form of the expression.

Simplification Steps

Breaking down an expression into its simplest form involves several steps, and understanding these empowers you to tackle complex algebraic tasks with ease. For the expression \(2x^0\), follow these steps to simplify:Start by identifying the exponent parts of the expression. In this case, it's \(x^0\). Knowing the Power of Zero rule, replace any such part with 1. After substitution, multiply it with other components of the expression.
Thus, \(2x^0\) becomes \(2\times1\). Finally, perform the multiplication to reach the simplest form, resulting in 2.
Breaking down the process using these simplification steps makes tackling even more challenging expressions doable and less intimidating. Always look to apply rules like the Power of Zero in contexts where your expression has potential simplifications.