Problem 28 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Check the results by squaring each root. $$ \pm \sqrt{900} $$

Step-by-Step Solution

Answer

The expression \(\pm \sqrt{900}\) evaluates to \(\pm 30\)

Step 1: Evaluate the square root

In this step, calculate the square root of 900, which is 30.

Step 2: Realize the square root sign represents both positive and negative

The squareroot sign \(\sqrt{}\) represents both the positive and the negative root of the number. So, the actual result of \(\pm \sqrt{900}\) is \(\pm 30\)

Step 3: Verification by squaring each root

To verify the results, square both 30 and -30. This should result same original number. \((30)^2 = 900\) and \((-30)^2 = 900\)

Key Concepts

Square Root Calculation

Understanding how to calculate square roots is fundamental in math. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, to calculate the square root of 900, you look for a number which, when multiplied by itself, equals 900. The square root of 900 is 30 because

\[ 30 \times 30 = 900 \]

This operation is often performed with the help of a calculator, but knowing how to calculate square roots manually, especially for perfect squares, can deepen your understanding of the concept.

Positive and Negative Roots

When working with square roots, it's important to recognize that every positive number has two square roots: one positive and one negative. This is because both a positive number and its negative counterpart, when squared, will result in the original positive number. For instance, both \( +30 \) and \( -30 \) are square roots of 900 since

\[ (+30)^2 = (+30) \times (+30) = 900 \]
and
\[ (-30)^2 = (-30) \times (-30) = 900 \]

Thus, when you see a symbol like \( \pm \sqrt{900} \) in an expression, it implies there are two potential square roots: \( +30 \) and \( -30 \) with the \( \pm \) indicating both.

Verification by Squaring

A critical part of working with square roots is the ability to verify your answers. Squaring the square roots that you've found is the best way to check your work. If squaring the result gives you the original number, your square root is correct. For the example of \( \pm \sqrt{900} \) where the square roots are \( \pm 30 \) the verification process involves squaring both 30 and -30.

\[ (30)^2 = 900 \]
and also
\[ (-30)^2 = 900 \]

If the verification fails, it means there was an error in the square root calculation. This step is crucial as it helps ensure the correctness of your solution.