Question

Find the value of each expression if \(x=2\) and \(y=-1\) \(\sqrt{x^{2}}+\sqrt{y^{2}}\)

Short answer

The value of the expression is 3.

Step-by-step solution

Substitute the values of x and y

First, substitute the values of x and y into the expression. Given that \( x = 2 \) and \( y = -1 \), the expression becomes: \( \sqrt{(2)^2} + \sqrt{(-1)^2} \).

Calculate the squares

Next, calculate the squares of the substituted values: \( \sqrt{4} + \sqrt{1} \).

Simplify the square roots

Now, simplify the square roots: \( 2 + 1 \).

Add the simplified values

Finally, add the simplified values together: 2 + 1 = 3.

Key concepts

Substitution Method

The substitution method is a technique used to solve problems by replacing variables with their given values. In our exercise, we are given specific values for the variables x and y. We then substitute these values into the expression to solve it. This step is crucial because it converts abstract expressions into simpler numerical calculations. For example, we replaced x with 2 and y with -1 to form the expression: \( \sqrt{(2)^2} + \sqrt{(-1)^2} \). Once we have the numerical values, we can proceed to solve the problem.

Square Roots

Square roots are a fundamental part of algebra. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2 because \( 2 \times 2 = 4 \). In our problem, we deal with two square roots: \( \sqrt{(2)^2} \) and \( \sqrt{(-1)^2} \). To find these square roots, first, calculate the squares: \( 2^2 = 4 \) and \( (-1)^2 = 1 \). Then, take the square roots of these results. \( \sqrt{4} = 2 \) and \( \sqrt{1} = 1 \).

Simplifying Expressions

Simplifying expressions is the process of breaking them down into their most basic form. After substituting and calculating the squares in our example, the next step is to simplify the square roots. Simplifying \( \sqrt{4} \) gives us 2, and simplifying \( \sqrt{1} \) gives us 1. Once we have these simplest forms, we can easily perform other operations, like addition, to find the final answer. Simplification makes complex expressions easier to work with and understand.

Addition

Addition is one of the basic operations in arithmetic. It combines two or more numbers into a single sum. In the final step of our example, we add the simplified values obtained from the square roots. So, we add 2 and 1 together: \( 2 + 1 = 3 \). This step is straightforward but essential. It provides the final answer to the problem after all the previous steps of substitution, square roots, and simplification have been completed.