Question

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

Short answer

2

Step-by-step solution

Substitute the Values

First, we need to substitute the values of the variables into the expression. Given that \(x = 2\) and \(y = -1\), we substitute \(x\) into the expression \((\sqrt{x})^{2}\). This gives us \((\sqrt{2})^{2}\).

Simplify the Expression

Next, we simplify the expression \((\sqrt{2})^{2}\). According to the properties of square roots, \((\sqrt{a})^{2} = a\). Therefore, \((\sqrt{2})^{2} = 2\).

Key concepts

Substitution in Algebra

In algebra, substitution is the process of replacing variables with their numerical values. It is a common technique used to solve equations and evaluate expressions. To perform substitution:

• Identify the variable(s) in the expression or equation.
• Replace each variable with the given value.
• Follow the order of operations to simplify the expression.

For example, in the expression \((\sqrt{x})^{2}\), given \(x = 2\), we substitute \(x\) with 2, resulting in \((\sqrt{2})^{2}\). Through substitution, we transform the algebraic expression into a numerical expression.

Simplifying Expressions

Simplifying an expression means reducing it to its simplest form. This can involve combining like terms, performing arithmetic operations, and applying mathematical properties. When simplifying:

• First substitute any variable values.
• Apply the order of operations (PEMDAS/BODMAS).
• Combine and reduce terms where possible.

In the example \((\sqrt{2})^{2}\), we recognize that the expression involves a square root squared. Using the property \((\sqrt{a})^{2} = a\), we simplify \((\sqrt{2})^{2}\) directly to 2. Simplification helps in making an expression more straightforward and easier to understand.

Properties of Square Roots

Square roots are mathematical functions that yield a number which, when multiplied by itself, results in the original number. Important properties of square roots include:

• The square root of a number squared returns the original number: \((\sqrt{a})^{2} = a\).
• Square roots are only defined for non-negative numbers in the real number system.
• The property \sqrt{ab} = \sqrt{a} \sqrt{b}\ can be useful when simplifying expressions.

Using these properties makes complex expressions easier to handle. For instance, in our problem, we use \((\sqrt{2})^{2} = 2\). Knowing these properties ensures accurate and efficient calculations in algebra.