Question

Evaluate the expression. \(b^{3}-5\) when \(b=2\)

Short answer

The evaluated result of the expression \(b^{3}-5\) when \(b=2\) is \(3\).

Step-by-step solution

Substitute the given value

Substitute the given value \(b=2\) into the expression \(b^{3}-5\). This gives us \(2^3 - 5\).

Simplify the expression

Simplify \(2^3 - 5\) to \(8 - 5\).

Perform the operation

Perform the operation \(8 - 5\) to give a final result of \(3\).

Key concepts

Substitution in Algebraic Expressions

Substitution is a fundamental method used in algebra that involves replacing variables with their respective numerical values. The objective is to transform an expression with variables into an arithmetic operation that can be easily evaluated.

For example, if you're given the expression \(b^3 - 5\) and instructed that \(b = 2\), the first step is to substitute \(2\) for every instance of \(b\). The expression then becomes \(2^3 - 5\), with no variables left, ready for further simplification.

Understanding substitution is essential because it's not only the initial step but also sets the stage for accurate calculations in later stages. Correctly substituting values ensures the foundation of solving the problem is sound.

Exponentiation in Algebra

Exponentiation is an operation that raises a number, known as the base, to a power, which is the exponent. It's a shorthand for repeated multiplication. In algebra, we frequently encounter expressions where variables are raised to exponents.

For instance, in \(2^3\), the base is \(2\) and the exponent is \(3\), indicating that \(2\) should be multiplied by itself two additional times: \(2 \times 2 \times 2\), resulting in \(8\). This operation is crucial for simplifying expressions and solving equations.

Understanding how to evaluate exponentiation correctly ensures that more complex algebraic expressions can be simplified accurately. This forms the basis for higher level algebra and is essential for grasping mathematical concepts that involve powers and roots.

Simplifying Algebraic Expressions

To simplify an algebraic expression means to make it as basic or reduced as possible. After substitution and evaluating any exponentiation, the expression might still need simplification. This involves performing all the operations, including multiplication, division, addition, or subtraction.

In the given example, after substituting \(2\) for \(b\) and evaluating \(2^3\), we get \(8 - 5\). Here, simplification involves doing the subtraction, which gives us \(3\). Always remember to follow the order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).

Grasping how to simplify algebraic expressions correctly is invaluable for solving algebra problems efficiently and is one of the essential skills in understanding and performing algebraic manipulation.