Problem 102 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(9^{0}\right)^{4}+\left(z^{0}\right)^{5} $$

Step-by-Step Solution

Answer

The simplified expression is 2.

Step 1: Apply Zero Exponent Rule

Recall that any non-zero number raised to the power of 0 equals 1. Thus,\[ 9^0 = 1 \]and\[ z^0 = 1 \] according to the zero exponent rule.

Step 2: Simplify Each Term

Now substitute these values back into the expression:\[ \left(9^0\right)^4 = 1^4 = 1 \]and\[ \left(z^0\right)^5 = 1^5 = 1 \].

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, called the base, to a specific power or exponent. This is expressed as \( a^n \), where \( a \) is the base and \( n \) is the exponent. The process is similar to repeated multiplication of the base by itself. For example:

• \( 3^2 \) means \( 3 \) multiplied by itself, so \( 3 \times 3 = 9 \).
• \( 4^3 \) indicates \( 4 \times 4 \times 4 = 64 \).

In the context of the zero exponent rule, any non-zero base raised to the power of zero is always 1, a fundamental principle in mathematics. This simplifies calculations involving zero exponents.
Understanding these concepts allows you to solve expressions like \( \, (9^0)^4 \), where each component under a zero exponent simplifies to 1.

Simplifying Expressions

Simplifying expressions is about making them easier to understand and work with—often by reducing them to their simplest form. When faced with complex algebraic expressions, follow these steps:

• Simplify exponents using rules such as the zero exponent rule.
• Perform operations like multiplication or addition as per the rules of arithmetic.
• Combine like terms and cancel out where possible.

As demonstrated in the original problem, simplifying means converting \( \, (9^0)^4 + (z^0)^5 \) to more manageable numbers. Initially, applying the zero exponent rule changes \( 9^0 \) and \( z^0 \) to 1. Then \( 1^4 \) and \( 1^5 \) both remain 1, making the whole expression equal to 2 after adding: \( 1 + 1 = 2 \).
This approach makes it clear how exponents function and how expressions can be simplified effectively.

Algebra Basics

Algebra forms the core of higher-level mathematics and revolves around finding the unknown values within an equation. Basic concepts include:

• Understanding variables, such as \( z \) in the expression.
• Utilizing arithmetic operations to manipulate expressions.
• Applying fundamental rules, like zero exponent and distribution laws.

Grasping these elementary principles is a stepping stone to more advanced algebraic operations. By knowing how zero exponents simplify variables and coefficients, you streamline solving equations and expand this understanding to tackle more complex algebraic problems.
Practicing problems like \( \, (9^0)^4 + (z^0)^5 \), helps reinforce these basics and elucidates how they underlie more intricate concepts in mathematics.