Problem 57 - Algebra 1

Question

Find all square roots of the number or write no square roots. Check the results by squaring each root. $$\frac{9}{25}$$

Step-by-Step Solution

Answer

The square roots of the number 9/25 are 3/5 and -3/5.

Step 1: Find square root of the numerator

The given fractional number is 9/25. We start by finding the square root of the numerator, which is 9. The square root of 9 is 3.

Step 2: Find square root of the denominator

Then move on to finding the square root of the denominator, which is 25. The square root of 25 is 5.

Step 3: Formulate the square roots

Now that we have the square roots of the numerator and the denominator, we can construct the square roots of the given fraction. We must remember that a number has two square roots, the positive and the negative one. So the square roots of 9/25 are 3/5 and -3/5.

Step 4: Verify the roots

We verify these roots by squaring each of them. Squaring 3/5 yields 9/25. Similarly, squaring -3/5 also yields 9/25. Therefore, our roots are correct.

Key Concepts

Fractional Numbers

Fractional numbers are mathematical expressions that represent a division of two integers.
The numerator is the top number, indicating how many parts we have, while the denominator is the bottom number, specifying into how many parts the whole is divided.
Understanding fractions is crucial because they often appear in problems involving square roots, as seen in the exercise with the fraction \(\frac{9}{25}\).

• To find the square root of a fraction, compute the square roots of both the numerator and the denominator separately.
• Reduce fractions to their simplest form if necessary before finding the square roots.

Both positive and negative square roots can exist for a fraction, just like whole numbers. Finding the square roots of fractions involves identifying these two possible values.

Positive and Negative Roots

Every positive number, including fractional numbers, has two square roots – one positive and one negative.
This duality arises because both a positive number and its negative counterpart square to give the same positive result.
For the fraction \(\frac{9}{25}\), both \(\frac{3}{5}\) and \(-\frac{3}{5}\) are square roots because:

• \(\left(\frac{3}{5}\right)^2 = \frac{9}{25}\)
• \(\left(-\frac{3}{5}\right)^2 = \frac{9}{25}\)

When solving problems, always consider both the positive and negative roots, unless the context of a problem limits to finding only positive square roots, such as in geometry.

Squaring Numbers

Squaring a number involves multiplying it by itself.
It is a fundamental operation in algebra and essential for verifying square roots.
For instance, to confirm \(\frac{3}{5}\) as a square root of \(\frac{9}{25}\), compute:

• Multiply \(\frac{3}{5}\) by itself: \(\frac{3}{5} \times \frac{3}{5} = \frac{9}{25}\).
• Similarly, \(-\frac{3}{5} \times -\frac{3}{5} = \frac{9}{25}\).

Squaring confirms that both \(\frac{3}{5}\) and \(-\frac{3}{5}\) are valid square roots. This helps ensure the correctness of solutions in exercises involving square roots.

Square Root Verification

Verifying square roots is crucial to ensure that the roots are correct.
This process involves squaring the potential roots to check if they result in the original number.
For \(\frac{9}{25}\):

• Calculate \(\left( \frac{3}{5} \right)^2\) to see if it equals \(\frac{9}{25}\).
• Similarly, compute \(\left( -\frac{3}{5} \right)^2\).
• Both should result in the original fraction, \(\frac{9}{25}\).

Verification clearly shows the extent to which square roots are accurate, removing errors in computation and achieving a stronger grasp of the concepts behind square roots.