Problem 23 - Algebra 1

Question

Find all square roots of the number or write no square roots. Check the results by squaring each root. $$49$$

Step-by-Step Solution

Answer

The square roots of 49 are 7 and -7.

Step 1: Find the positive square root

The square root of 49 can be determined by considering the whole numbers that multiply by themselves to give 49. In this case, the positive square root of 49 is 7 because \(7 \times 7 = 49\).

Step 2: Find the negative square root

Similarly, a number has a negative as well as a positive square root. That's because a negative number multiplied by another negative number yields a positive result. Hence, the negative square root of 49 is -7 (since \(-7 \times -7 = 49\).)

Step 3: Verify the results

It's always good to double-check the results. Consequently, square each root to ensure that it produces the original number. In this case, \(7^2 = 49\) and also \((-7)^2 = 49\). This confirms that both 7 and -7 are indeed the square roots of 49.

Key Concepts

Positive Square Root

When we refer to the 'positive square root', we're talking about the non-negative root of a given number. For instance, the positive square root of 49 is 7. Finding this root is akin to asking, 'What number can I multiply by itself to get 49?' To put it in mathematical terms, you'd write this as \( \sqrt{49} = 7 \). Remember, this root is the principal square root and is always non-negative.

Students often mistakenly assume there is only one square root because the positive one is more frequently used. However, including both roots gives us a complete understanding of square roots as a concept. When you're calculating the positive square root, ensure to consider positive answers only, as negative results indicate a different aspect of square roots.

Negative Square Root

In contrast to the positive square root, every positive number also has a 'negative square root'. This is because when a negative number is squared (multiplied by itself), the result is a positive number. For the number 49, its negative square root is -7, expressed as \( \sqrt{49} = -7 \). This idea is rooted in the property that \((-7) \times (-7) = 49\).

It's important to recognize that negative square roots are just as valid as their positive counterparts. They expand our understanding of numbers and their properties. When working with square roots, always remember to consider both the positive and negative square roots for a complete set of solutions.

Squaring Numbers

Squaring a number means multiplying that number by itself. The term 'square' comes from the geometric shape where all sides are of equal length — similar to how the same number is used on both sides of the multiplication. If you square the number 7, you calculate \(7 \times 7\), which equals 49. Mathematically, this is expressed as \(7^2 = 49\).

Squaring numbers is a fundamental operation that shows up in many areas of mathematics, including geometry and algebra. When squaring negative numbers, such as -7, the same principle applies, giving \((-7)^2 = 49\). Despite the negative sign, squaring always results in a positive product because a negative times a negative yields a positive.

Verifying Square Roots

Verifying square roots is an essential step in ensuring the accuracy of your mathematical calculations. To verify a square root, simply square it and see if the result matches the original number. For example, to check that 7 is a square root of 49, we calculate \(7^2\) and ensure the result is 49. Similarly, to verify -7 as a square root, we calculate \((-7)^2\) and confirm the result is also 49.

Verification is a powerful mathematical tool that proves the correctness of our solutions. It not only helps prevent errors but also deepens our understanding by reinforcing the relationships between squaring and square roots. Always verify both positive and negative roots to finalize your solutions confidently.