Problem 38 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Check the results by squaring each root. $$ \pm \sqrt{81} $$

Step-by-Step Solution

Answer

The square roots are \( +9 \) and \( -9 \)

Step 1: Evaluating the square root

The square root of a number \(x\) is a value that, when multiplied by itself, gives the original number \(x\). The square root of 81 is expressed as \( \sqrt{81} \). The square root of 81 is 9, i.e., \( \sqrt{81} = 9 \). However, the original problem included a ± symbol, which means we consider both the positive and negative square root. So, the roots are \( +9 \) and \( -9 \)

Step 2: Confirming the results by squaring each root

Now, we verify the roots by squaring them. Squaring means multiplying the number by itself. Hence, \( +9 * +9 = 81 \) and \( -9 * -9 = 81 \). So, by squaring, we obtain the original number from both positive and negative roots, confirming that the roots are correct.

Key Concepts

Square Root Calculation

Understanding how to calculate square roots is fundamental in mathematics. A square root of a number is a value that, when multiplied by itself, equals the original number. To find the square root, you essentially ask, 'What number multiplied by itself gives me this original number?' Let's apply this to an example: what is the square root of 81? The process involves finding a number that multiplied by itself equals 81. Through multiplication knowledge, or even using a calculator, you discover that 9 times 9 equates to 81. Therefore, you can say that \( \sqrt{81} = 9 \).

It's important to note that when you encounter a square root problem in mathematics, it is crucial to consider that there are standardized steps you should follow to arrive at the correct solution. This typically involves identifying the number you need the root of, and then applying basic multiplication facts or using tools like a calculator to find the square root.

Positive and Negative Square Roots

Many students initially learn that the square root of a number is positive. However, it's vital to recognize that numbers have both positive and negative square roots. This is because a negative number multiplied by itself also results in a positive number. For instance, both \( +9 \) and \( -9 \) are square roots of 81 because \( 9 \times 9 = 81 \) and \( -9 \times -9 = 81 \). When you see the symbol \( \pm \) before a square root, it indicates that you need to consider both square roots.

In mathematical notation, when we want to include both roots, we write \( \pm \sqrt{81} \), which means 'positive or negative square root of 81.' This should convey that two answers are required: \( +9 \) and \( -9 \). It is crucial for students to understand this concept to solve square root problems accurately.

Verifying Square Roots by Squaring

Once you've calculated the square roots, whether they are positive or negative, it's important to verify your results. How can you be sure that the roots you've found are indeed correct? The best way is to square the roots you’ve determined. Squaring is simply multiplying a number by itself, which should return you to the original number before taking its square root.

For example, after finding that the positive and negative square roots of 81 are \( +9 \) and \( -9 \), verify by squaring these numbers: \( 9 \times 9 = 81 \) and \( -9 \times -9 = 81 \). The fact that both operations give us back 81 confirms that \( \pm 9 \) are the correct square roots of 81. This step is a crucial part of the problem-solving process, ensuring that the roots are accurate, and it gives students confidence in their understanding of square roots.