Problem 16 - Algebra 2 and Trigonometry

Question

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ 4 x^{0} $$

Step-by-Step Solution

Answer

The value of the expression is 4 when \(x \neq 0\).

Step 1: Understanding Exponents

The expression given is \(4x^0\). To solve this, we first need to understand what \(x^0\) means. By the rules of exponents, any non-zero base raised to the power of zero equals 1. Therefore, \(x^0 = 1\) for any \(x eq 0\).

Step 2: Substituting the Exponent Result

Now that we know \(x^0 = 1\), substitute 1 into the expression \(4x^0\). This gives us \(4 \times 1\).

Step 3: Simplifying the Expression

Simplify the expression \(4 \times 1\) by performing the multiplication. The result is 4.

Key Concepts

Expression Simplification

Simplifying an expression means making it easier to understand or solve. It involves reducing the expression to its simplest form by combining like terms or using mathematical operations. In this exercise, we start with the expression \(4x^0\). Our goal is to break this down to a simpler form.

To do this, we use known mathematical rules and properties, such as exponent rules, to rewrite expressions more clearly. By recognizing that \(x^0 = 1\) for any non-zero \(x\), we can replace \(x^0\) with 1 in the expression. This eliminates the exponent and simplifies the expression to \(4 imes 1\), or just 4.

Simplifying expressions makes calculations neater and helps you to see the final value or solution more directly. It's like cleaning up a messy room to find what you're looking for more easily.

Exponent Rules

Understanding exponent rules is essential when working with expressions involving powers. Exponents tell you how many times to multiply a base by itself. A key exponent rule to remember is that any non-zero number raised to the power of zero equals one. This means that for any \(x eq 0\), \(x^0 = 1\).

Here are some important rules of exponents to keep in mind:

• \(a^0 = 1\) for any \(a eq 0\)
• \(a^m \times a^n = a^{m+n}\)
• \((a^m)^n = a^{m \times n}\)
• \(a^m / a^n = a^{m-n}\) when \(a eq 0\)

In our case, we used the rule \(x^0 = 1\) as part of simplifying the expression \(4x^0\). Seeing these rules in use helps reduce more complex expressions into standard forms that are easier to handle.

Multiplication of Expressions

Once you understand the role of numbers and variables in an expression, the next step is to perform operations such as multiplication. In mathematics, multiplication is a straightforward process of combining numbers or terms based on rules and properties.

In our initial expression \(4x^0\), once we simplify \(x^0\) to 1, the multiplication involves combining 4 and 1. It's a basic multiplication, but it serves as an essential step in simplifying expressions.

• Start by identifying each term that needs to be multiplied.
• Multiply the numerical coefficients or constants together.
• Multiply any remaining variables following the rules of exponents if applicable.

In this specific problem, since \(x^0\) equals 1, you simply calculate \(4 \times 1\), arriving at 4 as the simplified, final result. Such basic operations build a foundation for handling more complex algebraic expressions in the future.