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Chapter 10: Problem 9

Simplify each expression as completely as possible. Be sure your answers are in simplest radical form. Assume that all variables appearing under radical signs are nonnegative. $$\sqrt{400}$$

Short Answer

Expert verified
The simplest radical form of \(\text{sqrt}(400)\) is 20.

Step by step solution

01

- Identify Perfect Squares

Recognize that 400 is a perfect square. A perfect square is a number that can be expressed as the product of an integer with itself.
02

- Find the Square Root

Find the square root of 400. Observe that 400 can be expressed as \(20 \times 20\). Therefore, the square root \(\text{sqrt}(400) = 20\). This is because 20 is the number that, when multiplied by itself, gives 400.

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Key Concepts

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

Perfect Squares
A perfect square is a number that results from multiplying an integer by itself. For example, 1, 4, 9, 16, and 25 are all perfect squares.
  • 1 = 1 脳 1
  • 4 = 2 脳 2
  • 9 = 3 脳 3
  • 16 = 4 脳 4
  • 25 = 5 脳 5
Recognizing perfect squares can make simplifying square roots much easier. The original problem involves simplifying \(\root 400\).
We start by recognizing that 400 is a perfect square because there exists some integer, 20, such that 20 脳 20 = 400.
Square Root Calculation
Calculating the square root involves determining which number, when multiplied by itself, gives the original number. For example, \( \text{sqrt(25)} = 5 \).
To find the square root of 400, identify the number that, when multiplied by itself, equals 400. Notice that 20 脳 20 = 400. Therefore, the square root of 400 is 20.
\[\begin{equation} \text{sqrt}(400) = 20 \end{equation}\] This method works because the square root function is defined as the opposite of squaring a number.
Radical Expressions
A radical expression includes a radical sign (鈭�), which denotes the root of a number. When simplifying radical expressions, the goal is to express the number inside the radical in its simplest form.
Here's a quick checklist to simplify radical expressions:
  • Identify and factor the number inside the radical into its prime factors.
  • Group the factors into pairs when possible.
  • Move any complete pairs out from under the radical as a single number.
  • Multiply any remaining factors inside the radical.
In our example, the radical expression \(\root 400\) simplifies quickly because 400 is a perfect square.
Therefore, no further steps are needed beyond recognizing that \(\text{sqrt}(400) = 20\).

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Most popular questions from this chapter

Use a calculator to give decimal approximations rounded to one, two, and three places. $$\sqrt{17}$$

Reduce to lowest terms. $$\frac{4+\sqrt{8}}{6}$$

Use a calculator to give decimal approximations rounded to one, two, and three places. $$\sqrt{23}$$

Rationalize the denominators and simplify. $$(3+\sqrt{5})^{2}+\frac{8}{3-\sqrt{5}}$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{20}-\sqrt{5}$$
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