Problem 36 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$ b^{4} \text { when } b=9 $$

Step-by-Step Solution

Answer

The value of the expression \( b^{4} \) when \( b = 9 \) is \( 6561 \).

Step 1: Understanding the Problem

The problem is asking to calculate the value of the expression \( b^{4} \) when the value of the variable \( b \) is \( 9 \). Exponentiation is an arithmetic operation, in this case, with a base of \( b \) and an exponent of \( 4 \), which means \( b \) multiplied by itself three more times.

Step 2: Substituting the Value

Replace the \( b \) in \( b^{4} \) with \( 9 \), giving \( 9^{4} \).

Step 3: Calculating the Expression

Evaluate \( 9^{4} \), which means \( 9 \) multiplied by itself three more times. The result is \( 6561 \).

Key Concepts

Arithmetic operations

Arithmetic operations are the basic calculations you can perform with numbers, consisting of addition, subtraction, multiplication, division, and exponentiation. Exponentiation is a vital arithmetic operation where a number, called the base, is multiplied by itself a number of times indicated by its exponent. In the exercise, exponentiation is demonstrated through the expression \( b^4 \). This means that you need to take the base \( b \) and multiply it by itself a total of four times, or equivalently, do three repeated multiplications.

• The base is the number that is being multiplied.
• The exponent tells you how many times to use the base as a factor.

For example, \( 9^4 \) implies you multiply \( 9 \) by itself four times: \( 9 \times 9 \times 9 \times 9 \). Each instance where one number is multiplied by another is a step in a multi-part arithmetic problem. Understanding this operation is essential for working with powers and provides a foundation that is applicable in more complex equations.

Variables in algebra

In algebra, a variable is a symbol, usually a letter, that represents a number that can change or vary. Variables are used as placeholders for unknown values, and they help form expressions and equations. In the exercise, \( b \) is the variable used in the expression \( b^4 \).

• Variables allow you to write algebraic expressions that can apply to a wide range of values.
• When a specific value is assigned to a variable, the expressions can be evaluated, making computations possible.

Here, the problem specifies that the variable \( b \) should be replaced with \( 9 \). This substitution allows you to convert the algebraic expression into an arithmetic one, which can then be calculated. Variables thus play a crucial role in formulating problems and simplifying the computational process by allowing standard operations across different values.

Evaluating expressions

Evaluating an expression involves replacing the variable with a given number and calculating the result using proper arithmetic operations. In the current exercise, after substituting the value of \( b \) with \( 9 \), the next step is to evaluate \( 9^4 \).

• First, substitute the variable with its given value.
• Follow the order of operations to properly calculate the expression.

For instance, with \( 9^4 \), begin by calculating \( 9 \times 9 = 81 \), then \( 81 \times 9 = 729 \), and finally \( 729 \times 9 = 6561 \). Breaking the process down into multiple steps ensures you arrive at the correct solution.
Evaluating expressions is an essential part of algebra that lets you compute values, solve equations, and understand relationships between different variables. Recognizing this process helps in systematically tackling the expressions and attaining correct results.