Problem 64 - Algebra 1

Question

Evaluate the expression for the given value of the variable. (Review 1.2) $$(6 w)^{2} \text { when } w=5$$

Step-by-Step Solution

Answer

The evaluated result of the expression for \( w = 5 \) is 900.

Step 1: Substitution of the variable value

Substitute the given value of \( w = 5 \) into the expression \( (6w)^2 \). This substitution leads to \( (6*5)^2 \).

Step 2: Simplification of the expression

Perform the multiplication before squaring the result, according to BIDMAS. This gives \( 30^2 \).

Step 3: Final calculation

Square 30, which results in \( 30^2 = 900 \).

Key Concepts

Substitution

Understanding how to evaluate expressions often starts with substitution, which is when you replace a variable with its corresponding numerical value. For example, if you are given an expression like \( (6w)^2 \) and told that \( w = 5 \), the first step is to 'plug in' the value of 5 wherever you see a \( w \).

Think of it as a placeholder in a sentence that you fill in with the provided information. After the substitution, the expression will look like \( (6*5)^2 \), which no longer has the variable and is ready for the next step of solving the problem. Using substitution correctly is pivotal, as it sets the stage for all the subsequent calculations.

Simplifying Expressions

Once you've replaced the variables with numbers, the next thing on the agenda is simplifying expressions. Simplification refers to the process of making an expressions as straightforward as possible by performing all possible calculations. This could include operations such as multiplying numbers, combining like terms, or reducing fractions.

In our case, after substituting \( w \) with 5, we need to multiply 6 by 5 to simplify the expression. Simplifying is about breaking down complex equations into simpler parts in order to solve them more easily.

Order of Operations

To get to the correct answer when simplifying expressions, one must follow the order of operations. This is a fundamental concept that determines the sequence in which you should perform mathematical operations in an expression. The order of operations can be remembered by the acronym BIDMAS or PEMDAS, which stand for Brackets/Parentheses, Indices/Exponents, Division and Multiplication (left to right), Addition and Subtraction (left to right).

In evaluating \( (6*5)^2 \), according to BIDMAS, you must first do the multiplication inside the parentheses, giving you \( 30^2 \). Only after this do you move on to the next operation—in this case, squaring the number 30.

Squaring Numbers

Finally, the problem culminates in the operation of squaring numbers. Squaring is the process of multiplying a number by itself. Mathematically, if you have a number \( n \), squaring it is denoted by \( n^2 \), which is equal to \( n * n \).

For our exercise, after simplifying and using the order of operations, we square the number 30. So, \( 30^2 \) becomes \( 30 * 30 = 900 \). Squaring numbers is a basic arithmetic task, but it's important to be careful since larger numbers can lead to mistakes if you're not diligent. Remember, the square of a number is always positive, because multiplying two positive or two negative numbers both result in a positive number.