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Chapter 10: Problem 15

For each number, find all of its square roots. $$ 900 $$

Short Answer

Expert verified
The square roots of 900 are 30 and -30.

Step by step solution

01

Understand the Concept of Square Roots

Square roots are numbers that, when multiplied by themselves, result in the original number. For any positive number, there are two square roots: one positive and one negative.
02

Identify the Positive Square Root

To find the positive square root of 900, determine the number which when multiplied by itself equals 900. The positive square root of 900 is \(\frac{900}{2}/15=30\), as \(\frac{30 \times 30}{30}=900\).
03

Identify the Negative Square Root

For every positive square root, there is a corresponding negative square root. Therefore, the negative square root is \(\frac{-30}{-30}=-900\).
04

Verify the Results

Confirm by squaring both the positive and negative roots: \(\frac{30 \times 30}/900=900\) and \(\frac{-30 \times -30=900)/. Therefore, 30}/-900\) are both square roots of 900.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

positive square root
The square root of a number is a value that, when multiplied by itself, gives the original number. When we talk about the positive square root, we refer to the number without a negative sign. For instance, the positive square root of 900 is 30, since \(30 \times 30 = 900\).
The key thing to remember is:
  • Positive square roots are always non-negative numbers.
  • They are usually what calculators and most software return as the principal square root.
In the case of 900, the positive square root is clear and straightforward, \( \sqrt{900} = 30\). By understanding the positive square root, you begin to see how squaring and roots are inversely related.
negative square root
In addition to positive square roots, every positive number also has a negative square root. This is simply the negative version of the positive square root. For example, for the number 900, the negative square root is -30 because \(-30 \times -30 = 900\).
Here's what you should keep in mind:
  • Negative square roots are simply the negative counterparts of positive square roots.
  • They are essential because they provide a complete set of solutions to the square root equation.
Always remember, for any positive number, the negative root is just as valid:
\( -\sqrt{900} = -30\). Understanding negative square roots helps in solving equations and understanding symmetry in mathematics.
verifying square roots
Verifying square roots involves squaring the potential roots to ensure they produce the original number. It's an essential step to confirm your solutions are accurate. To verify the square roots of 900:
  • Square the positive root: \(30 \times 30 = 900\).
  • Square the negative root: \(-30 \times -30 = 900\).
Both results must return the original number.
This process can be summarized as:
1. Identify the potential roots.
2. Multiply the identified roots by themselves.
3. Check if the resultant equals the original number.
If both calculations hold true, you have correctly identified both square roots. Verification is a dependable way to ensure the accuracy of your solutions.

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