Problem 51 - Algebra 1

Question

Find all square roots of the number or write no square roots. Check the results by squaring each root. $$64$$

Step-by-Step Solution

Answer

The square roots of 64 are 8 and -8.

Step 1: Identify the Square Root

The square root of a number \( n \) is the value that, when multiplied by itself, gives the original number \( n \). To find the square root of 64, you can use the square root operation. The positive square root of 64 is 8 because \( 8 \times 8 = 64 \).

Step 2: Identify the Negative Square Root

All positive real numbers have two square roots. Apart from the positive square root, there is also negative square root. The negative square root of 64 is -8 because \( -8 \times -8 = 64 \).

Step 3: Check the Results

To confirm that the values obtained are correct, check the results by squaring each root. \( 8^2 = 64 \) and \( (-8)^2 = 64 \). Both results are accurate since they give back the original number from which the roots were derived.

Key Concepts

Positive Square Roots

Square roots are interesting because they deal with finding a number that, when multiplied by itself, returns to the original number. When we discuss positive square roots, we are identifying the non-negative number that satisfies this condition.
For instance, the positive square root of 64 is 8. This is because multiplying 8 by itself results in 64.
If you take any positive number like 64, it will always have a positive square root. These are often marked by the radical symbol (√), so the positive square root of 64 is denoted by \( \sqrt{64} \), which equals 8.

Negative Square Roots

While it might seem like square roots are all about positive numbers, there's more to the story. Any positive number actually has two square roots—one positive and one negative.
The negative square root is the opposite of its positive counterpart. For our example with 64, while 8 is the positive square root, -8 serves as the negative square root.
This is because \( (-8) \times (-8) = 64 \), as multiplying two negative numbers results in a positive product. So, negative square roots are valid and can be found by just taking the negative of the positive square root.

Checking Square Roots

After calculating square roots, whether positive or negative, it is prudent to verify your answers. Checking square roots involves squaring the root to see if it returns to the original number.
Take the time to do this step, as it ensures the accuracy of your calculations.

• For a positive root like 8, verify by calculating \( 8^2 = 64 \).
• For a negative root like -8, verify by calculating \( (-8)^2 = 64 \).

This method is foolproof because if squaring the root gives you back to the initial number, then the root is correct.