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Chapter 10: Problem 32

Evaluate or simplify each expression. (Assume \(x \geq 0 .)\) $$(\sqrt{x})^{2}$$

Short Answer

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Step by step solution

01

Understand the Concept of a Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. The square root of x is denoted by \( \sqrt{x} \) and is defined as the number that satisfies \( ( \sqrt{x} \) )^2 = x.\GIVEN: \( ( \sqrt{x} \) )^2 \
02

Apply the Definition

According to the definition of a square root, raising \( \sqrt{x} \) to the power of 2 will give back the original number x.
03

Simplify the Expression

Simplify the expression \( ( \sqrt{x} \) )^2. Based on the definition, \( ( \sqrt{x} \) )^2 = x. Therefore, the simplified form is x.

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Key Concepts

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

simplifying expressions
Simplifying expressions is a fundamental aspect of algebra. It involves reducing complex expressions to simpler or more manageable forms without changing their value. Simplification makes equations easier to understand and solve.

In our given exercise, we simplify the expression \(( \sqrt{x} )^{2}\).

  • Start by understanding each part of the expression.
  • Recognize components, such as square roots or exponents.
  • Apply relevant algebraic rules to combine or reduce them.

In this case, we use the property of square roots and exponents. Specifically, squaring a square root returns the original value. The simplified form here is 'x'.
power of a number
The 'power of a number', or exponent, is a basic concept in algebra. It signifies how many times a number, called the base, is multiplied by itself. It's written as \( base^{exponent} \).

In the exercise, the given expression is \( ( \sqrt{x} )^2.\)

  • The base here is \( \sqrt{x} \).
  • The exponent is 2.
  • The expression means \( \sqrt{x} \) multiplied by itself.

According to the property of exponents, \( (a^m)^n = a^{m*n} \). Applying this, we get \( x^{1/2*2} = x^{1} = x. \)

Understanding exponents helps us simplify and evaluate complex algebraic expressions with ease.
elementary algebra
Elementary algebra is the branch of mathematics dealing with basic algebraic concepts. It includes working with variables, understanding expressions, and solving simple equations.

In the given instance, we work with a variable 'x' and its square root. Here鈥檚 a quick recap:

  • The concept of a variable: 'x' represents an unknown or any value.
  • Square roots: \( \sqrt{x} \) represents a number which, when squared, returns 'x'.
  • Exponents: Here, the exponent '2' implies squaring the base.

Combining these, \( ( \sqrt{x} )^{2} = x. \) This demonstrates how basic algebra manipulates and simplifies expressions. Knowing these basics is essential for tackling more complex algebraic problems.

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Most popular questions from this chapter

Estimate the given square root between two consecutive integers without using a calculator, then use a calculator to find the square root rounded to two decimal places to confirm your estimate. $$\sqrt{673}$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$8 \sqrt{25}-3 \sqrt{21}$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$x \sqrt{28 x y^{3}}+y \sqrt{63 x^{3} y}$$

Estimate the given square root between two consecutive integers without using a calculator, then use a calculator to find the square root rounded to two decimal places to confirm your estimate. $$\sqrt{73}$$

Perform the indicated operations. Simplify all answers as completely as possible. Assume that all variables appearing under radical signs are nonnegative. $$\frac{10}{4-\sqrt{11}}$$
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