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Chapter 0: Problem 104

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

Short Answer

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Step by step solution

01

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 \((\sqrt{2})^{2}\).
02

Simplify the Expression

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{x+h}-\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}$$

Simplify each expression. $$25^{3 / 2}$$

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{3}-1}{2 \sqrt{3}+3}$$

Rationalize the numerator of each expression. Assume that all variables are positive when they appear. $$\frac{4-\sqrt{x-9}}{x-25} x \neq 25$$

Simplify each expression. $$4^{3 / 2}$$
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