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Chapter 7: Problem 78

Why is it incorrect to say that, in general, \(\sqrt{x^{2}}=x ?\)

Short Answer

Expert verified
The correct relationship is \( \sqrt{x^{2}} = |x| \/\), not \( x \/\), because \( \sqrt{x} \/\) returns only the principal (non-negative) square root.

Step by step solution

01

Understand the Square Root and Squaring Function

Recognize that squaring a number and taking the square root are inverse operations, but this relationship depends on the value of the number.
02

Consider Positive and Negative Values

Acknowledge that squaring a number (whether positive or negative) always results in a non-negative value.
03

Analyze the Principal Square Root

Understand that the square root function, \( \sqrt{x} \/\), returns only the principal (non-negative) square root of a number.
04

Apply the Properties to the Equation

Recognize that when \( \sqrt{x^{2}} \/\) is taken, the result should always be non-negative, while \( x \/\) can be either positive, negative, or zero.
05

Conclusively State the Correct Relationship

The correct relationship is \( \sqrt{x^{2}} = |x| \/\), where \( |x| \/\) denotes the absolute value of \( x \/\), ensuring the result is always non-negative.

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Key Concepts

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

Square Root Function
The square root function is a special mathematical operation that finds a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 times 3 equals 9. This is written as: \[ \sqrt{9} = 3 \] Keep in mind that every positive number has two square roots: one positive and one negative. This is because both 3 and -3, when squared, equal 9. However, the square root function, by convention, returns only the non-negative result, known as the principal square root.
  • It is represented as \sqrt{x}.
  • Always gives a non-negative result.
  • For any non-negative number x, \sqrt{x}*\sqrt{x} = x.
This is an important concept to understand, as it directly affects other operations and properties related to square roots.
Inverse Operations
Inverse operations are mathematical operations that reverse the effect of each other. Squaring a number and taking the square root are inverse operations. For example: \[ (\sqrt{9})^{2} = 9 \] and \[ \sqrt{(3)^{2}} = 3 \]
When you square a number, you multiply it by itself. Taking the square root is the opposite 鈥� you find the number that, when squared, gives you the original value. It's essential to note:
  • Squaring a positive or negative number results in a positive number or zero.
  • Square roots must deal with non-negative numbers because the principal square root only considers non-negative values.
Remember, these inverse operations are crucial for solving equations and understanding various mathematical properties.
Principal Square Root
The principal square root is the non-negative square root of a number. For any number x: \[ \sqrt{x} \] it only returns a value that is zero or positive.
This concept is vital because it sets the standard for how we interpret and calculate square roots. For example, while \pm 3 both satisfy that \pm 3 * \pm 3 = 9, \sqrt{9} will always be 3 under the principal square root.
  • Ensures that \sqrt{x^{2}} always equals |x|.
  • The absolute value is used to guarantee a non-negative result.
  • Allows consistent and clear interpretation, especially when solving equations.
Using the principal square root simplifies many mathematical processes and provides a consistent approach to calculations involving square roots.

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Most popular questions from this chapter

Simplify. $$i^{2}+i^{4}$$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt[4]{a^{2} b}(\sqrt[3]{a^{2} b}-\sqrt[5]{a^{2} b^{2}}) $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt[4]{9 a b^{3}} \sqrt{3 a^{4} b} $$

Simplify. $$ \frac{1}{2} \sqrt{36 a^{5} b c^{4}}-\frac{1}{2} \sqrt[3]{64 a^{4} b c^{6}}+\frac{1}{6} \sqrt{144 a^{3} b c^{6}} $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt{2 x^{3} y^{3}} \sqrt[3]{4 x y^{2}} $$
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