Problem 25 - College Algebra

Question

Simplify each exponential expression $$ x^{0} y^{5} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(x^{0}*y^{5}\) is \(1*y^{5}\) or simply \(y^{5}\).

Step 1: Simplify \(x^0\)

Any non-zero number to the power of 0 equals 1. So, \(x^0 = 1\)

Step 2: Simplify \(y^5\)

\(y^5\) is already simplified, as 5 is a prime number.

Step 3: Write the simplified form of the expression

Combine the simplified forms of \(x^0\) and \(y^5\)

Key Concepts

Zero Exponent Rule

In algebra, the zero exponent rule is a fundamental concept that simplifies equations. When any non-zero number is raised to the power of zero, it equals 1. This might seem counterintuitive at first, but it's a useful rule that helps in simplifying expressions.
The logic behind this is based on the properties of exponents. For example:

• Consider the expression \( x^n \div x^n \). Using exponent rules, this simplifies to \( x^{n-n} = x^0 \).
• We also know that any number divided by itself is 1, hence \( x^n \div x^n = 1 \).

Both results lead us to conclude that \( x^0 = 1 \). Always remember that this rule only applies when the base \( x \) is not zero.

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their simplest form. This includes combining like terms and utilizing exponent rules such as the zero exponent rule.
When you simplify, you are transforming the expression into one that is easy to understand and less prone to calculation errors. Let's see how this works:

• Take \( x^0 y^5 \) as an example: Since \( x^0 = 1 \), the expression simplifies to \( 1 \cdot y^5 \).
• This results in \( y^5 \), which is the simplest form of the original expression.

This step-by-step approach ensures clarity and accuracy when working with more complex algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They don't have an equal sign, unlike equations, and can often appear complicated. However, mastering the art of dealing with algebraic expressions is key in mathematics.
Consider this:

• An expression such as \( x^0 y^5 \) consists of the variable \( y \) raised to the power of 5 and the variable \( x \) raised to the power of zero.
• The simplification process, as seen before, leads to \( y^5 \). In this context, understanding each component's role, like the zero exponent, is crucial.

By breaking down expressions and applying rules systematically, you gain better insight and become more adept at manipulating and simplifying algebraic expressions.