Question

Evaluate the exponential expression. \(v^{3}+w\) when \(v=3\) and \(w=5\)

Short answer

The evaluated result of the given expression is 32.

Step-by-step solution

Identify the values

Given, v = 3 and w = 5.

Calculate v^3

Since v = 3, we have \(v^{3} = 3^{3} = 27\)

Evaluate v^3 + w

Finally, to evaluate \(v^{3}+w\), substitute \(v^{3}\) and w into the equation. This gives us \(27+5 = 32\).

Key concepts

Exponents

Exponents are a fundamental concept in algebra that describe how many times a number, known as the base, is multiplied by itself. For instance, the exponential expression \( 3^3 \) means that the base \( 3 \) is used as a factor three times. In simpler terms, \( 3 \times 3 \times 3 \) equals 27.

Understanding how to work with exponents is crucial in algebra, as they appear in various types of problems, from simplifying algebraic expressions to solving complex equations. A key rule to remember with exponents is that when a base is raised to the power of one, the result is the base itself, and when raised to the power of zero, the result is always one, regardless of the base value (except for the base zero).

When evaluating expressions with exponents, it's important to perform the exponentiation before any addition or subtraction, according to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Substitution Method

The substitution method is an approach used to evaluate algebraic expressions. It involves replacing variables with their corresponding numerical values. This method is integral to simplifying expressions and solving equations.

To effectively use the substitution method, you must:

• Identify the values of the variables given in the problem. In the case of our exercise, \( v = 3 \) and \( w = 5 \) are the given values.
• Proceed by replacing the variables in the expression with the numbers you've identified. This means wherever you see a \( v \) in the expression, you would put a 3, and similarly, replace a \( w \) with a 5.
• Carefully perform the arithmetic operation that the expression dictates.

By following these steps, you can simplify expressions or solve equations in a systematic and error-free manner.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. Unlike equations, these expressions don’t contain an equality sign. They are used to represent relationships and to formulate equations that can be solved.

For example, the algebraic expression in our exercise, \( v^{3}+w \) , can represent a wide range of values depending on what \( v \) and \( w \) are. This expression is composed of two parts: \( v^{3} \) which shows an exponentiation, and \( w \) which is a variable. Both parts are connected by the addition operation.

In algebra, understanding how to interpret and manipulate these expressions is key to solving problems. Practice helps to make the process of working with algebraic expressions feel more intuitive over time. It's equally important to become comfortable transitioning between verbal, numerical, and algebraic expressions as this skill can provide a deeper understanding of the concepts at hand.