Problem 38 - Algebra 1

Question

Evaluate the expression. Give the exact value if possible. Otherwise, approximate to the nearest hundredth. $$\pm \sqrt{6.25}$$

Step-by-Step Solution

Answer

\(\pm 2.5\)

Step 1: Interpret the \(\pm\) symbol

The \(\pm\) symbol means that the function could output a positive or a negative number. For a square root calculation, this indicates that the result could be either a positive root or a negative root.

Step 2: Calculate square root

Square root of 6.25 is calculated. The exact square root of 6.25 is 2.5.

Step 3: Apply \(\pm\) to the result

Apply the \(\pm\) symbol to the result. This gives us \(\pm 2.5\), meaning the result could be either 2.5 or -2.5.

Key Concepts

Integer Operations

When working with square roots, it is important to have a basic understanding of integer operations. These are the simple calculations involving whole numbers, which include addition, subtraction, multiplication, and division. In the context of square roots, we often deal with integers before and after obtaining the root.

• Addition and subtraction are frequently used to simplify expressions before calculating the square root. For example, if you have a number like 9 + 16 under a square root symbol, you would first perform the addition to get 25 before finding its square root, which is 5.
• Multiplication and division help to simplify or rearrange expressions, particularly in complex equations that include square roots. For instance, if you have \( \sqrt{36} \, * \,\ \sqrt{9} \, \), compute the square roots first. \( \sqrt{36} = 6 \) and \( \sqrt{9} = 3 \), then multiply to get 18.

These integer operations are foundational skills that make it easier to estimate and manipulate numbers, especially when dealing with square roots.

Approximation

In mathematics, not every square root has a neat, exact integer solution. This is where approximation comes into play. Approximation is the process of finding a number that is close enough to the true value of a square root for practical purposes. It is especially useful for square roots of non-perfect squares.

• Using approximation helps simplify calculations when an exact square root is difficult or time-consuming to find. You estimate the root value to a certain degree of accuracy, such as the nearest hundredth or tenth depending on the requirement.
• When you approximate, you might use tools like calculators or estimate by noting that \( \sqrt{70} \) is between \( \sqrt{64} \) (8) and \( \sqrt{81} \) (9), so it is going to be a number between 8 and 9, more specifically around 8.37.

These small estimates can greatly aid in both academic and real-world applications where precision is crucial but the exact square root is not necessary.

Positive and Negative Numbers

Positive and negative numbers are a crucial aspect when dealing with the \( \pm \) symbol, especially in square roots. The \( \pm \) symbol indicates that both the positive and negative square roots should be considered.

• Positive numbers are values greater than zero. When you take the square root of a number, the principal square root is typically the positive root. For example, the principal square root of 6.25 is 2.5.
• Negative numbers are values less than zero. When involved with the \( \pm \) symbol, the negative root must also be taken into account; hence, for 6.25 and the use of \( \pm \, \sqrt{6.25} \, \), we get -2.5 as another potential value.

Understanding how positive and negative numbers work in conjunction with roots helps avoid mistakes when solving equations or expressions that involve \( \pm \) symbols. It ensures that both possible solutions are acknowledged in contexts where both are relevant, such as quadratic equations.