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Chapter 0: Problem 45

Find each square root. If it is not a real number, say so. $$ -\sqrt{400} $$

Short Answer

Expert verified
-20

Step by step solution

01

Understand the Problem

The given expression is \(-\sqrt{400}\). The goal is to find the square root of 400 first and then consider the negative sign.
02

Calculate the Square Root

Calculate the square root of 400. The square root of a number is a value that, when multiplied by itself, gives the original number. Since \(20 \times 20 = 400\), \(\sqrt{400} = 20\).
03

Apply the Negative Sign

Place the negative sign in front of the square root value found in Step 2. Hence, \(-\sqrt{400} = -20\).
04

Consider Real Number Property

Since \(-20\) is a valid real number, this expression does not produce an imaginary or complex result.

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Key Concepts

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

real numbers
Real numbers include all the numbers that you can find on the number line. This category encompasses both positive and negative numbers, fractions, and irrational numbers too. When we talk about real numbers, we鈥檙e discussing numbers you can measure or count in the real world. For example, 2, -3.5, and \(\frac{7}{2}\) are all real numbers. They are directly opposed to imaginary numbers, which cannot be located on the number line. When you鈥檙e dealing with square roots, the result will be a real number as long as you don't attempt to take the square root of a negative number. This is because the square root of a negative number is not a real number but an imaginary one.
negative square root
A negative square root involves taking the square root of a number and then applying a negative sign. This is different from taking the square root of a negative number, which does not produce a real number. For instance, considering our problem \(-\sqrt{400}\), we first find the square root of 400. The square root of 400 is 20, and then we apply the negative sign, giving us -20. It's important to understand that while the result is negative, the process involves finding the square root of a positive number first and then applying the negative. This ensures that the outcome is a real number.
simplifying square roots
Simplifying square roots means turning a square root into its simplest form. For perfect squares, simplification is straightforward. Importantly, a perfect square is an integer that is the square of another integer. For example: \(400 = 20 \times 20\), so \(\sqrt{400} = 20\). When simplifying non-perfect squares, you look for factors that are perfect squares. For example, to simplify \(\sqrt{50}\): split 50 into \(\sqrt{25 \times 2}\). Since 25 is a perfect square, you extract its square root, simplifying to \(\sqrt{25} \times \sqrt{2} = 5 \sqrt{2}\). This method allows you to simplify more complex square roots step-by-step by breaking them down into more manageable parts.

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