Problem 18 - Elementary and Intermediate Algebra

Question

Simplify each expression. \(8 t^{0}\)

Step-by-Step Solution

Answer

The simplified expression is 8.

Step 1: Understanding Zero Exponents

Any number raised to the zero power, except zero itself, is equal to one. This means that if you have an expression like \(t^{0}\), it simplifies directly to 1.

Step 2: Applying Zero Exponent Rule

Since \(t^{0} = 1\), we can replace \(t^{0}\) with 1 in the expression \(8 t^{0}\). Thus, it simplifies to \(8 \times 1\).

Step 3: Simplifying the Multiplication

Now multiply 8 by 1. Any number multiplied by 1 remains unchanged, so \(8 \times 1 = 8\).

Key Concepts

Understanding Exponents

Exponents are an important part of mathematics, especially in algebra. They are used to denote repeated multiplication of the same number or expression. Exponents consist of a base and a power, with the power indicating how many times the base is multiplied by itself. For instance, in the expression \(2^3\), 2 is the base, and 3 is the exponent or power, which means \(2 \times 2 \times 2\).

One special case to note is the zero exponent rule. According to this rule, any non-zero number raised to the power of zero equals one, expressed as \(a^0 = 1\) where \(a eq 0\). This rule simplifies expressions and makes calculations easier, avoiding undefined results that might occur if zero is used as a base.

Simplifying Expressions

Simplifying expressions helps in reducing complexity and making calculations straightforward. To simplify an expression means to combine like terms and reduce it to its most basic form.

When an expression involves an exponent, as in the case of zero exponents, simplification becomes essential. For example, when simplifying \(8 t^0\), we apply the zero exponent rule to recognize \(t^0\) as 1. This reduces our expression to \(8 \times 1 = 8\). Simplification ensures that we present the expression as clearly and concisely as possible.

To simplify any expression:

• Identify terms that can be combined (like terms)
• Apply multiplication or division rules as needed
• Use exponent rules effectively to simplify power terms

These steps help in reducing algebraic expressions to their simplest form, making them easier to work with.

An Introduction to Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like addition and multiplication). These expressions are a fundamental part of algebra, forming the basis for equations and functions.

An algebraic expression can contain:

• Variables (letters that represent numbers, e.g., \(t, x, y\))
• Constants (specific numbers, e.g., 8, -5)
• Operators (symbols that denote mathematical operations, such as \(+, -, \times, \div\))

For instance, in the expression \(8 t^0\), 8 is the constant, \(t^0\) involves a variable \(t\) with an exponent, and the product represents multiplication.

Understanding algebraic expressions is fundamental for solving equations, modeling real-life scenarios, and performing advanced calculations. Simplifying these expressions often involves using the rules of exponents, applying arithmetic to constants, and combining like terms to streamline and solve them more easily.