Problem 36 - Intermediate Algebra

Question

In the following exercises, evaluate the following expressions. When \(a=3, b=8\) \(a^{2}+b^{2}\)

Step-by-Step Solution

Answer

The value of the expression is 73.

Step 1: Identify given values

Identify the values given in the problem: \(a=3\) and \(b=8\).

Step 2: Understand the expression

Understand that you need to evaluate the expression \(a^{2}+b^{2}\).

Step 3: Compute square of a

Calculate \(a^{2}\) by substituting \(a=3\): \(3^2 = 9\).

Step 4: Compute square of b

Calculate \(b^{2}\) by substituting \(b=8\): \(8^2 = 64\).

Step 5: Add the results

Add the results from the previous steps: \(9 + 64 = 73\).

Key Concepts

algebraic expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operation symbols. Variables like 'a' and 'b' are symbols that represent numbers, while operations include addition, subtraction, multiplication, and division. Algebraic expressions can be as simple as a single variable or a sum of complex terms. In this exercise, the expression given is a^2 + b^2.
This means you require squaring the variables 'a' and 'b' and then adding the results. Algebraic expressions are fundamental in algebra and help in forming equations and solving problems involving unknowns.

substitution method

The substitution method is a technique used to simplify and solve algebraic expressions. Here, we need to substitute the given values of the variables into the expression. Given that a=3 and b=8, we replace 'a' and 'b' in a^2 + b^2 with these values.
It then becomes 3^2 + 8^2. This method allows us to work with numbers instead of variables, making evaluations straightforward. The substitution method is fundamental for solving equations and evaluating expressions in algebra.

square of a number

The square of a number means multiplying the number by itself. Mathematically, if xequals a number, its square is represented as x^2. In our exercise, we compute the squares of 3 and 8. To find 3^2, we calculate 3 * 3 = 9. Similarly for 8, 8^2 becomes 8 * 8 = 64. Squaring is a basic operation in algebra and is often used to solve quadratic equations and other mathematical problems. Knowing how to square numbers quickly is important for efficiency in problem-solving.

step-by-step problem solving

Step-by-step problem solving is an effective strategy to tackle mathematical problems systematically. It involves breaking down the problem into smaller, manageable steps.
In this exercise, we followed these steps:
• **Step 1:** Identify the given values: a=3 and b=8.
• **Step 2:** Understand the expression to be evaluated: a^2 + b^2.
• **Step 3:** Compute the square of a.
• **Step 4:** Compute the square of b.
• **Step 5:** Add the results from the previous steps to get the final answer: 9 + 64 = 73.
This method not only helps in achieving the correct result but also in understanding each part of the problem deeply. It builds a strong foundation for solving more complex algebraic problems in the future.