Problem 29 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for then given value of the variable. \(b^{3}\) when \(b=9\)

Step-by-Step Solution

Answer

The evaluated expression \(b^{3}\) when \(b = 9\) equals \(729\).

Step 1: Understand the expression

First, understand that \(b^{3}\) means \(b \times b \times b\). It is the cube of the variable \(b\).

Step 2: Substitute the value of the variable

Substitute the given value \(b = 9\) into the expression. So, the expression becomes \(9^{3}\) or \(9 \times 9 \times 9\).

Step 3: Evaluate the expression

Perform the multiplication operation. Multiplying \(9 \times 9 \times 9\) gives us \(729\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent tells us how many times we multiply the base by itself. For example, when we see an expression like \(b^{3}\), this means that we should multiply the base, \(b\), by itself three times: \(b \times b \times b\).

This process can be applied to any base raised to any positive integer exponent, and the resulting number is much larger than the base itself when the exponent is greater than one. It's a shorthand way of representing repeated multiplication and is particularly useful when dealing with large numbers.

Substituting Values

Substituting values into an algebraic expression is a fundamental skill in algebra. The process involves replacing the variable in the expression with the given number or another expression. By doing so, you can simplify and evaluate the expression to find its numerical value.

For instance, if you have the expression \(b^{3}\) and you know that \(b = 9\), then you can substitute 9 for \(b\) everywhere it appears in the expression. After substitution, the expression would look like \(9^{3}\) which can then be evaluated. It's essential to follow the proper order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), when performing the evaluation.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. Expressions can represent general relationships between quantities and can be simplified or evaluated when variable values are known.

In our example, \(b^{3}\) is an algebraic expression which describes the cube of \(b\). These expressions become especially useful when solving equations, modelling real-world scenarios, and finding patterns in numbers. Algebraic expressions do not have an equals sign; when an equals sign is involved, it becomes an equation.

Multiplication

Multiplication is one of the basic arithmetic operations and indicates repeated addition. In algebra, multiplication involving variables allows us to rewrite lengthy sums compactly and perform calculations more efficiently.

When we evaluate \(b^{3}\) with \(b = 9\), we multiply 9 by itself three times, which is \(9 \times 9 \times 9\). This process can be visualized as stacking layers or adding groups of the same number repeatedly. The result of multiplication is called the product. In this case, the product of \(9 \times 9 \times 9\) is 729.