Problem 28 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for then given value of the variable. \(w^{2}\) when \(w=12\)

Step-by-Step Solution

Answer

The evaluated value of the expression \(w^{2}\) when \(w=12\) is 144.

Step 1: Substitute the given value

In the given expression \(w^{2}\), replace the variable \(w\) with the given value 12. After substitution, the expression becomes \(12^{2}\).

Step 2: Simplify the expression

Calculate the expression \(12^{2}\) by multiplying 12 by itself, which is 144.

Key Concepts

Substitution

Substitution is like a simple switcheroo, where you replace a variable in an expression with a number or another value. In the world of math, it's crucial for evaluating expressions, especially when given specific values. For instance, if you have an expression like \(w^2\) as in the example, and you're told \(w = 12\), you need to substitute the \(w\) with 12. This means wherever you see \(w\), you'll now write 12. So, \(w^2\) becomes \(12^2\). Substitution helps us to find numerical values for expressions that have variables. It involves literally plugging in the given value into the expression, which then allows us to move on to the next step: simplifying the expression.

Simplifying Expressions

Simplifying expressions involves breaking down complex expressions into simpler forms. After substitution, we often have mathematical operations to perform. The aim is to reach the simplest form of the mathematical expression. In the example \(12^2\), simplifying means performing the multiplication \(12 \times 12\). Simplification isn’t just about doing arithmetic, though. It includes combining like terms and reducing fractions in more intricate expressions. But for \(12^2\), it's a straightforward calculation which gives us an answer of 144. Simplifying makes the expression as easy and concise as it can be, allowing us to identify the precise value or simpler form of the expression.

Exponents

Exponents can seem a little tricky at first, but they are essentially a shorthand way to show repetitive multiplication. When we see something like \(w^2\), it means \(w\) is multiplied by itself once. The small number or "power", here 2, tells us how many times to multiply the base number, which is \(w\), by itself. In the example, \(w = 12\), so \(12^2\) means multiply 12 by 12. When dealing with exponents, remember:

• The base is the number being multiplied.
• The exponent tells us how many times to multiply the base by itself.

Knowing how to work with exponents is fundamental as they frequently appear in expressions and equations, making calculations more manageable by condensing multiplying operations into a more compact form.