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Chapter 8: Problem 51

Why is there no real number equal to \(\sqrt{-64}\) ?

Short Answer

Expert verified
There is no real number equal to \(\root{-64}\) because the square of a real number cannot be negative.

Step by step solution

01

Understanding Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, \(\root{4}\)=\root{16}\root{鈥}=4 because \root{4} multiplied by \root{4} equals 16.
02

Analyzing Negative Numbers

Recall that the square of any real number, whether positive or negative, is always positive. For example, \(3^2 = 9\) and \((-3)^2 = 9\).
03

Applying to Negative Radical

Since the square of any real number (positive or negative) is always positive, there cannot be a real number that, when squared, results in a negative number like \(-64\).
04

Conclusion

Thus, the square root of a negative number, such as \(\root{-64}\), is not a real number because no real number squared equals a negative value.

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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 on the number line, including both positive and negative integers, fractions, and irrational numbers. Some examples of real numbers are 3, -1, 0.5, and \(\root{2}\). Real numbers are crucial in everyday calculations and measurements. When we refer to real numbers, we mean any number that doesn't include an imaginary component.
Square Roots
The square root of a number is a value that, when multiplied by itself, produces the original number. For instance, \(\root{4}\) because \root{4x4} = 16\root{4x4} = 16 means the square root of 16 is 4. Square roots are represented using the radical symbol \(\root{}\). Keep in mind that all non-negative numbers have two square roots, a positive (called the principal square root) and a negative root.
Negative Numbers
Negative numbers are the numbers less than zero and are often represented with a minus sign (e.g., -5, -2.3). An important property of negative numbers is that when you square them, the result is always positive. For example, \( \sqrt{}(-3) ^2 = (-3) x (-3) = 9\), meaning \( -3^2 = 9\). Thus, squaring any negative number results in a positive product.
Imaginary Numbers
Imaginary numbers are used to represent the square roots of negative numbers. The fundamental unit of imaginary numbers is denoted by 'i,' where \(i^2 = -1\). Thus, for the given problem of \(\root{-64}\), no real number can solve it, but we can express it using imaginary numbers. The square root of \(-64\) can be written as \(8i\), since \(8i^2 = 8(-1) = -64\). Imaginary numbers extend the real number system to the complex number system, allowing us to solve equations involving the square roots of negative numbers.

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Most popular questions from this chapter

Vince wants to make a square patio in his yard. He has enough concrete to pave an area of 130 square feet. Use the formula \(s=\sqrt{A}\) to find the length of each side of his patio. Round your answer to the nearest tenth of a foot.

Simplify. \(\frac{6}{3-\sqrt{7}}\)

Simplify. (a) \(\sqrt{-49}\) (b) \(-\sqrt{256}\)

Simplify. (a) \(\sqrt{144}\) (b) \(-\sqrt{121}\)

Simplify. (a) \(\sqrt{-121}\) (b) \(-\sqrt{289}\)
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