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Chapter 1: Problem 103

Should \(\sqrt{9}\) be evaluated as 3 or \(\pm 3\) ?

Short Answer

Expert verified
\(\sqrt{9} = 3\), not \(\pm 3\).

Step by step solution

01

Understanding the Square Root

The square root of a non-negative number, like 9, refers to the non-negative number which, when multiplied by itself, gives the original number (in this case, 9). This is known as the principal square root.
02

Calculating the Principal Square Root of 9

To find the principal square root of 9, determine the positive number which, when squared, gives 9. The number is 3 because \(3^2 = 9\).
03

Exploring 卤3: Misconception Clarification

The expression \(\pm 3\) represents two numbers, 3 and -3, both of which are solutions to the equation \(x^2 = 9\). However, \(\sqrt{9}\) is asking specifically for the principal square root, which is just the positive number.
04

Final Decision

Hence, the square root symbol \(\sqrt{}\) only represents the non-negative root, making \(\sqrt{9} = 3\), not \(\pm 3\).

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Key Concepts

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

Principal Square Root
When we talk about the square root of a number, it's crucial to understand the concept of the "principal square root." The principal square root is the non-negative root of a non-negative number. Think of it as the "standard" root we typically refer to when using the square root symbol \( \sqrt{} \). This is crucial because it creates a clear cut way of interpreting square roots without ambiguity.
  • It is always positive or zero.
  • It simplifies calculations and comparisons by consistently providing a single, except zero, positive value.
The principal square root is the foundation for solving many practical and theoretical problems in mathematics.
Non-negative Number
A non-negative number is a number that is either positive or zero. Understanding non-negative numbers is deeply connected to the principal square root because the square root operation is defined for non-negative numbers only.
  • All whole numbers, including zero, are non-negative numbers.
  • Fractions and decimals greater than or equal to zero are also non-negative numbers.
This concept is important because it helps students understand which numbers are eligible to have real square roots without delving into complex numbers. By sticking to non-negative numbers, we ensure the results of our square root calculations are within the realm of real, easily understandable, values.
Misconception in Mathematics
One common misconception in mathematics occurs when students interpret \( \sqrt{9} \) as \( \pm 3 \). This arises from a misunderstanding of the principal square root concept. While both 3 and -3 satisfy the equation \( x^2 = 9 \), only the non-negative root, 3, is considered as \( \sqrt{9} \). This specific root is the principal square root, not both.
  • The principal square root symbol \( \sqrt{ } \) inherently excludes negative numbers.
  • Misunderstanding this concept can lead to mistakes in algebra and calculus.
Correctly distinguishing between principal square roots and general solutions to quadratic equations is fundamental to mathematical literacy and aids in avoiding errors in more complex problems.

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Most popular questions from this chapter

BUSINESS: Break-Even Points and Maximum Profit City and Country Cycles finds that if it sells \(x\) racing bicycles per month, its costs will be \(C(x)=420 x+72,000\) and its revenue will be \(R(x)=-3 x^{2}+1800 x\) (both in dollars). a. Find the store's break-even points. b. Find the number of bicycles that will maximize profit, and the maximum profit.

ATHLETICS: Juggling If you toss a ball \(h\) feet straight up, it will return to your hand after \(T(h)=0.5 \sqrt{h}\) seconds. This leads to the juggler's dilemma: Juggling more balls means tossing them higher. However, the square root in the above formula means that tossing them twice as high does not gain twice as much time, but only \(\sqrt{2} \approx 1.4\) times as much time. Because of this, there is a limit to the number of balls that a person can juggle, which seems to be about ten. Use this formula to find: a. How long will a ball spend in the air if it is tossed to a height of 4 feet? 8 feet? b. How high must it be tossed to spend 2 seconds in the air? 3 seconds in the air?

True or False: \(x=3\) is a vertical line.

A 5 -foot-long ramp is to have a slope of \(0.75\). How high should the upper end be elevated above the lower end? [Hint: Draw a picture.]

Can the graph of a function have more than one \(x\) intercept? Can it have more than one \(y\) -intercept?
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