Problem 67 - Algebra 1

Question

Evaluate the expression for the given value of the variable. (Review 1.2) $$5 x^{2} \text { when } x=16$$

Step-by-Step Solution

Answer

The evaluated value of the expression when x=16 is 1280.

Step 1: Identifying the Given

The given expression is \(5x^{2}\) and the value of x provided is 16.

Step 2: Substituting the Value of x

Now, substitute the value of x into the given expression. It will be like: \(5 (16)^{2}\).

Step 3: Simplifying the Expression

Next, carry out the exponentiation operation. After that, perform the multiplication. Hence, the simplified expression would be \(5 \times 256= 1280\).

Key Concepts

The Substitution Method

When tackling algebraic expressions, the substitution method is a fundamental skill to grasp. It's like a puzzle where you have all the pieces, but you need to put them in the right places to see the whole picture. Simply put, it involves replacing variables with their given numeric values.

Let's apply this method to our example. You are told to evaluate the expression 5x^2 for x = 16. Initially, you'll have an expression with an unknown quantity, x. The substitution method asks you to take the provided number, which is 16 in this case, and place it wherever x appears. This transforms 5x^2 into 5(16)^2, setting the stage for further simplification.

Keep in mind, the substitution must be done carefully, ensuring numbers take the exact place of the variables and that the original structure of the expression is maintained. This avoids any mix-ups in subsequent steps of solving the equation.

Exponentiation

Exponentiation is when you raise a number to the power of another. It sums up repeated multiplication in a neat, compact form. For instance, the phrase, 'two raised to the third power' is expressed as 2^3 and equals 2 * 2 * 2 = 8.

In our example, after the substitution method, we encounter (16)^2. This means we must multiply 16 by itself. Hence, 16 times 16 equals 256. This part often confuses students, so remember, the exponent tells you how many times to use the base as a multiplier. It does not mean to multiply the base by the exponent. The distinction is crucial for accurate calculation.

Note that with higher numbers or larger exponents, the calculations increase in complexity, often requiring a calculator. However, understanding the logic behind exponentiation is essential before relying on digital tools.

Simplifying Expressions

The final step is to simplify the expression. Simplifying can be thought of as cleaning up a messy room so everything's easy to find. After substituting and exponentiating, we need to perform the operations in the correct order; this follows the rules of the arithmetic hierarchy.

In our example, you're left with 5 x 256 after substitution and exponentiation. Multiplication is our final task to simplify this to a single number. Multiply 5 by 256 to get 1280, presenting a tidy, simplified answer.

Simplifying expressions not only makes them easier to understand but is crucial in finding the right solution to an algebra problem. Just like cleaning that messy room, it's about doing things one step at a time – order naturally leads to clarity.