Problem 11 - Prealgebra

Question

List all square roots of the given number. If the number has no square roots, write “none”. ?144

Step-by-Step Solution

Answer

The square roots of 144 are 12 and -12.

Step 1: Understand the Problem

The exercise asks us to find all the numbers that, when squared, result in 144. This means we need to find both the positive and negative square roots of 144.

Step 2: Determine the Positive Square Root

To find the positive square root of 144, we need to find the number that multiplies by itself to give 144. By calculating, we find that 12 multiplied by 12 equals 144, so one square root of 144 is 12.

Step 3: Determine the Negative Square Root

Recall that a negative number squared also results in a positive number. Therefore, -12 multiplied by -12 also equals 144, meaning the negative square root of 144 is -12.

Step 4: List All Square Roots

Now that we've found both the positive and negative square roots, we can list them. The square roots of 144 are 12 and -12.

Key Concepts

Positive Square Root

The concept of a positive square root is fundamental in mathematics, especially when trying to solve problems that involve square roots. Every non-negative number has a principal square root, commonly referred to as the positive square root. Essentially, if you take a positive square root of a number, you're looking for a positive number that, when multiplied by itself, gives the original number.
For example, with the number 144, you find that the positive square root is 12, because:

• 12 x 12 = 144

Identifying the positive square root is straightforward: compute or use a calculator if it's not readily evident. Positive square roots are often denoted by the radical symbol, \,\( \sqrt{\cdot} \,\), without any sign, indicating the principal or positive root. For 144, this symbolically means:

• \( \sqrt{144} = 12 \)

Negative Square Root

While the positive square root is the most commonly used, understanding the negative square root is equally important, especially in solving algebraic equations and certain mathematical models. The negative square root of a number is simply the negative of the positive square root. It involves finding a negative number that, when squared, yields the original positive number.
For the number 144, the negative square root is found as follows:

• -12 x -12 = 144

This leads us to conclude that the negative square root of 144 is -12. Negative square roots are crucial when considering solutions to quadratic equations, where typically both positive and negative roots are part of the solution set. Generally expressed with a negative sign in front, it combines with the square root symbol, for example:

• \( -\sqrt{144} = -12 \)

Understanding Square Roots

Grasping the concept of square roots goes beyond rote memorization—it involves understanding why we have both positive and negative square roots and their applications. A square root of a number is a value that, when multiplied by itself, gives that number. This is why every positive number technically has two square roots: one positive and one negative.
The basic principles to remember include:

• Square roots are defined for non-negative numbers and result in real numbers.
• Negative numbers do not have real square roots; instead, they have complex roots.

The understanding can be deepened by remembering that square roots undo the process of squaring. For instance, since squaring either 12 or -12 results in 144, the square roots of 144 are both 12 and -12. This dual nature reflects the inherent symmetry in squaring and extracting roots.
Applications of square roots span from solving quadratic equations in algebra to calculating dimensions in geometry, where recognizing both roots helps in considering all potential solutions or scenarios.