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Chapter 7: Problem 13

Find all of its square roots. $$ 400 $$

Short Answer

Expert verified
The square roots of 400 are 20 and -20.

Step by step solution

01

- Understand Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. Mathematically, if \( x \) is a square root of \( n \), then \( x^2 = n \).
02

- Identify the Positive Square Root

For the given number 400, we need to find \( x \) such that \( x^2 = 400 \). To find the positive square root: \[ x = \sqrt{400} \]. We know that 20 multiplied by 20 equals 400, hence \( \sqrt{400} = 20 \).
03

- Identify the Negative Square Root

For every positive square root, there is also a negative counterpart. Therefore, if \( 20 \) is a square root of 400, \( -20 \) will also be a square root because \( (-20)^2 = 400 \).
04

- List All Square Roots

The two square roots of 400 are 20 and -20.

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Key Concepts

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

positive and negative square roots
When we find the square root of a number, such as 400, we normally look for two roots: one positive and one negative. This is because both a positive and a negative value, when squared, give the original number. For example, both 20 and -20 are square roots of 400 because:
  • 20 脳 20 = 400
  • -20 脳 -20 = 400
It is important to always remember to include both the positive and negative square roots when solving such problems. This is why the complete answer for the square root of 400 includes both 20 and -20.
properties of square roots
Square roots have several interesting properties that are helpful when solving mathematical problems:
  • Non-negative Principal Square Root: The principal square root refers to the non-negative square root of a number. For 400, the principal square root is 20.
  • Symmetry: Square roots are symmetric with respect to zero. This means that for every positive square root, such as 20, there is a corresponding negative square root, -20.
  • Multiplicative Property: The square root of a product is the product of the square roots. For example, \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\). This property makes it easier to simplify complex square root expressions.
  • Square of a Square Root: The square of the square root of a number gives the original number back, i.e., \( (\sqrt{n})^2 = n\).
Understanding these properties can greatly simplify your calculations and improve your problem-solving skills.
finding square roots
Finding the square root of a number involves determining which value multiplied by itself equals the original number. For instance, we found the square root of 400 as follows:
  • Decomposition Method: Break down 400 into its factors, such as 20 脳 20. Here, it鈥檚 easy to see that both factors are 20, so one square root is 20. The other square root is -20 because \(( -20 )^2 = 400\).
  • Using a Calculator: A simple and quick way is to use a calculator to find the square root. Most calculators have a square root function that will yield the principal (positive) square root, which in this case is 20.
  • Estimation Method: This involves estimating by finding two closest square numbers. For 400, since it鈥檚 between 324 (18^2) and 441 (21^2), we narrow it down to 20. Confirming, we get 20.
Using these different methods can help reinforce your understanding and ensure that you can find square roots in various contexts.

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Most popular questions from this chapter

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