Problem 23 - Elementary and Intermediate Algebra

Question

Write each expression in an equivalent form using an exponent. $$ \frac{t}{2} \cdot \frac{t}{2} \cdot \frac{t}{2} $$

Step-by-Step Solution

Answer

The expression is \( \left(\frac{t}{2}\right)^3 \).

Step 1: Identify repeated multiplication

The expression given is \( \frac{t}{2} \cdot \frac{t}{2} \cdot \frac{t}{2} \). Notice that the term \( \frac{t}{2} \) is repeated three times, indicating repeated multiplication.

Step 2: Use properties of exponents

When a term is multiplied by itself multiple times, it can be expressed using exponents. The expression \( \frac{t}{2} \cdot \frac{t}{2} \cdot \frac{t}{2} \) can be rewritten as \( \left(\frac{t}{2}\right)^3 \) because \( \frac{t}{2} \) is repeated three times.

Key Concepts

Repeated Multiplication

Repeated multiplication occurs when the same number or algebraic expression is multiplied by itself multiple times. This form of multiplication is common in mathematics and can be found in many equations and formulas.
When you see something like \( a \cdot a \cdot a \), it means the number \( a \) is multiplied by itself three times. Instead of writing it out multiple times, we can use a shorthand notation called exponents.

• For example, \( a \times a \times a = a^3 \).
• Here, the exponent is 3, indicating that the base \( a \) is used in the multiplication three times.

The use of exponents not only simplifies writing, but also makes mental calculations easier and helps in understanding larger equations typically found in algebra.

Properties of Exponents

The properties of exponents are rules that simplify expressions involving repeated multiplication.
They are indispensable when working with exponents because they provide a shortcut for performing calculations.
Here are some key properties of exponents you should know:

• Product of Powers Property: \( a^m \cdot a^n = a^{m+n} \). This property allows you to add the exponents when multiplying like bases.
• Power of a Power Property: \( (a^m)^n = a^{m \cdot n} \). This helps simplify expressions where an exponentiated term is raised to another power.
• Power of a Product Property: \( (a \cdot b)^n = a^n \cdot b^n \). This is used when a product is raised to an exponent.

In the given exercise, instead of writing \( \frac{t}{2} \) three times, we use \( \left( \frac{t}{2} \right)^3 \). This simplifies the expression according to the properties of exponents.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division.
They can represent values and quantities that are not fixed, allowing for a general formulation of mathematical problems.
In an algebraic expression, variables act as placeholders and can assume different values.

• For example, in our exercise, \( \frac{t}{2} \) represents a fraction where \( t \) is a variable.
• When we write \( \left( \frac{t}{2} \right)^3 \), the expression becomes an exponentiated form of the original problem, indicating the algebraic nature of its solution.

Understanding algebraic expressions helps you manipulate and simplify equations, solve for unknowns, and apply mathematical logic in practical and theoretical scenarios.