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Chapter 10: Problem 11

Find all square roots of each number. $$400$$

Short Answer

Expert verified
The square roots of 400 are \(卤20\), which include the positive square root, 20, and the negative square root, -20.

Step by step solution

01

Identify the positive square root

Since we are looking for a number that, when multiplied by itself, gives the original number (400), you can determine the positive square root of 400 as follows: \(\sqrt{400} = 20\) This is because 20 times 20 equals 400: \(20 \times 20 = 400\)
02

Identify the negative square root

Since the negative square root is the negative counterpart to the positive square root, we simply add a negative sign: \(-\sqrt{400} = -20\) Therefore, we have two square roots of the number 400, which are: Positive square root: 20 Negative square root: -20

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Key Concepts

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

Positive Square Root
When we talk about the positive square root of a number, we are referring to the number which, when multiplied by itself, will return the original number. For example, in the case of 400, the positive square root is 20. This is because if we multiply 20 by itself, we get 400, which verifies our result.
  • Positive square roots are always non-negative numbers.
  • Every positive number has exactly one positive square root.
It is important to remember that the symbol for the square root, \( \sqrt{} \), always represents the positive square root unless a negative sign is explicitly included.
Negative Square Root
Contrary to positive square roots, negative square roots involve finding the negative counterpart of the positive square root. For example, if the positive square root of 400 is 20, then its negative square root would be -20.
  • Negative signs are used to indicate this square root.
  • Just like positive square roots, negative square roots are multiplied by themselves to give the original number, but negative.
This means if you multiply -20 by -20, you'll still get 400, completing the pair of square roots.
Problem Solving
Square roots are a common problem-solving element in mathematics, especially in algebra. The challenge typically involves determining what number, when squared, results in the given number.
When approaching a problem involving square roots, follow these steps:
  • Identify if you need both positive and negative roots.
  • Calculate the positive square root first.
  • Simply add a negative sign to get the negative square root.
Understanding square roots is crucial for solving quadratic equations and other algebraic expressions, as knowing these roots can reveal possible solutions for these expressions.
Algebraic Operations
Square roots play an essential role in various algebraic operations and expressions. Recognizing when to use square roots can simplify complex equations and help in insights into quadratic functions.
Square roots are often used in:
  • Simplifying expressions.
  • Solving quadratic equations.
  • Understanding the properties of numbers.
In algebra, the knowledge of square roots allows you to perform operations like factoring and completing the square, techniques that are often covered after mastering basic square root concepts.

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