Question

What is the value of the expression \(g \cdot(g+1)^{2}\) for \(g=2 ?\) F. 10 G. 12 H. 18 J. 20 K. 36

Short answer

Question: Find the value of the expression g . (g+1)^2 when g = 2. Answer: H. 18

Step-by-step solution

Substitute the value of g into the expression

Replace g with 2 in the expression: (2) . (2+1)^2.

Calculate inside the parentheses

Next, we need to evaluate the expression inside the parentheses: (2+1) = 3.

Evaluate the expression with the updated value

Now, rewrite the expression using the value found in step 2: (2) . (3)^2.

Calculate the square of 3

We need to find the square of 3: (3)^2 = 9.

Evaluate the complete expression

Finally, rewrite the expression using the value found in step 4: (2) . (9) = 18.

Match the answer with the given options

Our final answer is 18, which matches with option H, so the correct choice is H. 18.

Key concepts

Algebraic Expressions

Understanding algebraic expressions is crucial to mastering ACT Math Preparation. An algebraic expression consists of numbers, variables, and operations. These components combine to represent a specific value or set of values. When dealing with an algebraic expression, such as the one in our exercise, \(g \cdot(g+1)^{2}\), we observe that it involves a variable \(g\), an operation of multiplication, and an exponentiation.

For a better grasp, let’s consider algebraic expressions as recipes. A recipe tells you what ingredients you need and how to combine them to make a dish. Similarly, an algebraic expression tells you what numbers and variables you need and how to perform operations on them to find your answer. When you substitute a value for the variable \(g\), you're essentially choosing your ingredient to create the final product—the value of the expression.

Substitution Method

The substitution method is a fundamental tool in algebra which allows us to find the value of expressions by replacing variables with actual numbers. This is especially helpful when solving equations or evaluating expressions as seen in the provided exercise.

Why is this so important? Substitution is akin to testing a hypothesis with real data. By plugging in the value of \(g=2\), we move from a theoretical expression to concrete computation. To use this method effectively, replace the variable with the given number and ensure all instances of the variable in the expression are substituted. After substituting, you simply perform the operations as you normally would with numbers. It's a methodical process that connects abstract algebraic concepts to concrete values, and that's where the magic happens!

Exponentiation

Exponentiation is an operation that involves raising a number, known as the base, to a power, which is the exponent. It's a shorthand for multiplying the base by itself as many times as indicated by the exponent. In our ACT Math Preparation exercise, exponentiation is shown as \(3)^{2}\), meaning 3 is the base and 2 is the exponent.

Understanding this concept is critical because it often appears in various forms of mathematical problems. Always remember that the exponent corresponds to the number of times you multiply the base by itself. For example, \(3)^{2} = 3 \times 3 = 9\). Exponentiation can greatly change the result of an expression, making it a powerful tool. Consequently, mastering exponents enables you to simplify and evaluate expressions swiftly and helps in revealing patterns within numbers.