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Chapter 8: Problem 64

Explain why \(\sqrt{x}=-1\) has no solution.

Short Answer

Expert verified
The equation \(\sqrt{x} = -1\) has no solution because the principal (positive) square root of any real number cannot be negative by definition.

Step by step solution

01

Understanding Square Roots

The square root of a number finds a value that, when multiplied by itself, gives the original number. This is inherently a positive operation, since any number multiplied by itself gives a positive number, or zero.
02

Applying the Square Root Property

The square root function \(\sqrt{x}\) is defined such that it always returns a non-negative number. In other words, for any real number \(x\), \(\sqrt{x} \geq 0\). This is also known as the principal square root, and it is the positive square root of a number.
03

Evaluating \(\sqrt{x} = -1\)

If we go by the definition of square root, it's clear that \(\sqrt{x} = -1\) cannot have a solution because the square root of a number cannot be negative. Therefore, the equation \(\sqrt{x} = -1\) implies a negative square root, which contradicts the definition of the square root function.

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Most popular questions from this chapter

Square the real number \(\frac{2}{\sqrt{3}} .\) Observe that the radical is eliminated from the denominator. Explain whether this process is equivalent to rationalizing the denominator.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$25^{-\frac{1}{2}}=-5$$

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$243^{-\frac{1}{5}}$$

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Make Sense? In Exercises \(90-93,\) determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I use the fact that 1 is the multiplicative identity to both rationalize denominators and rewrite rational expressions with a common denominator.
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