Problem 59 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Give the exact value if possible. Otherwise, approximate to the nearest hundredth. $$ \pm \sqrt{1} $$

Step-by-Step Solution

Answer

The solutions are 1 and -1.

Step 1: Understanding the ± symbol

The symbol \(\pm\) means plus and minus. Therefore, when you see \(\pm\sqrt{1}\), there will be two possible solutions, one positive and one negative.

Step 2: Calculate the square root of 1

The square root of 1 is 1. This is because 1*1 = 1.

Step 3: Apply the ± symbol

Now, apply the ± symbol to the calculated square root. So, there will be two answers, +1 (the positive square root of 1) and -1 (the negative square root of 1).

Key Concepts

Understanding the Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. It's symbolized by a radical sign \( \sqrt{} \). For example, the square root of 4 is 2 because \( 2 \times 2 = 4 \). When evaluating the square root of a positive number, there is always a positive root, known as the principal square root. However, for every positive number except zero, there is also a negative root since \( -a \times -a = a^2 \).

In the case of zero, which is neither positive nor negative, the square root is also zero. This is a unique case because \( 0 \times 0 = 0 \). Understanding this concept is crucial because it forms the basis of solving equations and understanding higher-level math concepts involving radicals.

The Plus and Minus Symbol

The \( \pm \) symbol represents two possibilities: addition and subtraction. When this symbol precedes a number or an expression, it indicates that both the positive and negative values of the subsequent number or expression are to be considered. This dual nature is vital in mathematics, particularly in solving quadratic equations, where two possible solutions arise from the nature of squaring. In our example, \( \pm \sqrt{1} \) translates to two potential answers: +1 and -1.

To effectively deal with the \( \pm \) sign in mathematical expressions, one must break it down into its two separate scenarios and solve for each one individually. This ensures that all possible solutions are accounted for, thus providing a complete understanding of the problem's scope.

Exact Value Versus Approximation

In mathematics, we often distinguish between the 'exact value' of an expression and its 'approximated value'. An exact value is a precise number that has not been rounded or altered, whereas an approximation is a value that has been rounded to a certain number of decimal places or significant figures, often to make the number easier to work with. For instance, the exact value of \( \sqrt{2} \) cannot be expressed as a simple fraction or decimal because it is an irrational number, so we might approximate it to 1.41. However, some square roots result in whole numbers and can be stated exactly, such as \( \sqrt{1} = 1 \) or \( \sqrt{4} = 2 \).

Recognizing when a number can be described exactly or when it should be approximated is a key skill in mathematics. This concept not only applies to square roots but to other areas of math, such as trigonometry and logarithms, where exact values can sometimes be attained through known ratios and properties.