Problem 47 - Algebra 1

Question

Evaluate the expression. \(12+y^{3}\) when \(y=3\)

Step-by-Step Solution

Answer

The evaluated expression is 39.

Step 1: Identify the given variable

Here, the variable in the algebraic expression is \(y\) and it is given that \(y=3\).

Step 2: Substitute the value into the expression

Replace \(y\) in the expression \(12 + y^3\) by \(3\) to obtain \(12 + 3^3\).

Step 3: Carry out the operations

First compute the cube of 3 to get \(3^3 = 27\). Then, add this result to 12 to evaluate the entire expression: \(12 + 27 = 39\).

Key Concepts

Substitution

When working with algebraic expressions, substitution is a fundamental tool. It involves replacing the variable, often represented by a letter, with a given numerical value. In our exercise, the expression is \(12 + y^3\), and we're told that \(y = 3\). The substitution here is simply swapping the letter \(y\) with the number 3. This turns the expression into a straightforward numerical computation: \(12 + 3^3\).

• Identify what the variable represents.
• Ensure that you have the correct numerical value given for the variable.
• Replace the variable with this number in the expression.

By mastering substitution, you lay a strong groundwork for solving more complex algebraic problems.

Exponents

Exponents are a way to represent repeated multiplication. When we see \(y^3\), it means \(y\) is multiplied by itself twice more: \(y \times y \times y\). In our example, after substituting \(y\) with 3, we calculate \(3^3\), which is the same as \(3 \times 3 \times 3\). Computing this gives 27.

• An exponent represents how many times to multiply the base number by itself.
• In \(a^n\), \(a\) is the base and \(n\) is the exponent.
• Practice with different numbers to get familiar with the concept.

Understanding exponents is crucial for evaluating expressions where variables are raised to powers.

Order of Operations

The order of operations is a set of rules that determines the sequence in which calculations are performed. It is critical in ensuring that everyone solves algebraic expressions consistently and accurately. The rules can be remembered by the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (ie powers and roots, etc.)
• M/D: Multiplication and Division (left-to-right)
• A/S: Addition and Subtraction (left-to-right)

For the expression \(12 + 3^3\), we handle the exponent first, calculating \(3^3\) to get 27. Then we add 12, coming to the final result of 39.

By strictly following the order of operations, you ensure that mathematical expressions are evaluated correctly and without ambiguity.