Problem 79 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \(\frac{45}{a^{2}}\) when \(a=2\)

Step-by-Step Solution

Answer

The solution to the expression \(\frac{45}{a^{2}}\) when \(a=2\) is \(11.25\).

Step 1: Substitute value into the expression

Replace \(a\) in the expression \(\frac{45}{a^{2}}\) by \(2\). So, we get \(\frac{45}{2^{2}}\).

Step 2: Simplify the expression

Simplify \(2^{2}\) to \(4\), then \(\frac{45}{2^{2}}\) becomes \(\frac{45}{4}\).

Step 3: Complete the Calculation

To find the solution, divide \(45\) by \(4\). It equals \(11.25\).

Key Concepts

Understanding Substitution

Substitution is a key mathematical operation where you replace a variable with a specific value. This process is crucial for evaluating expressions. In the exercise given, the variable is \(a\), which is replaced by the value \(2\). Doing so transforms the expression from \(\frac{45}{a^2}\) to \(\frac{45}{2^2}\). Substitution helps us move from an abstract expression to one that can be directly calculated.

• Identify the variable in the expression.
• Insert the given value in place of the variable.
• Simplify the transformed expression to move towards finding the solution.

This foundational step ensures that the expression accurately represents the problem you are solving. Remember, substitution can be used in a variety of mathematical contexts, making it a versatile tool for solving equations.

Simplification after Substitution

Once substitution is completed, the next crucial step in expression evaluation is simplification. Simplification involves reducing expressions into their simplest form. After substituting \(a = 2\) in our example, we have \(\frac{45}{2^2}\). The expression inside the denominator, \(2^2\), simplifies to \(4\).

Simplifying the expression ensures that calculations are as straightforward as possible. Here are the simplification steps:

• Compute any exponents in the expression. For instance, \(2^2 = 4\).
• Replace the exponentiated number with its result in the expression.
• Transition to the next operation with a clearer expression.

By simplifying, you make subsequent arithmetic operations easier and less prone to error. If you simplify correctly, you're set up nicely for the next step: division.

Executing Division

After substitution and simplification, division is the final operation that solves the exercise expression. Division can initially feel intimidating, but breaking it down step by step makes it manageable.

In our expression \(\frac{45}{4}\), we divide \(45\) by \(4\). This gives us the final result of \(11.25\). Here's a simple process to keep in mind:

• Ensure the dividend (the number on top) and the divisor (number below) are ready for division. In this case, \(45\) and \(4\) respectively.
• Perform the division either by hand or using a calculator for precision.
• When working with decimals, aim for clarity and accuracy in your result.

Mastering division not only helps you solve expressions like these but also builds a foundation for tackling more complex problems in mathematics. With the correct division, you've evaluated the expression successfully.