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Chapter 8: Problem 52

Explain how you can tell that the equation \(\sqrt{x}=-8\) has no real number solution without performing any algebraic steps.

Short Answer

Expert verified
The square root function cannot equal a negative number, so \(\backslashsqrt{x} = -8 \ \) has no real solution.

Step by step solution

01

Understanding the Square Root Function

The square root function \(\backslashsqrt{x}\) represents a non-negative value for any real number \ x \. This means that the result of \(\backslashsqrt{x}\) is always zero or positive, never negative.
02

Analyze the Given Equation

The equation given is \(\backslashsqrt{x} = -8\). This shows that a negative value exists on the right-hand side of the equation.
03

Conclusion Based on Properties

Since the square root function cannot produce a negative value, we conclude that \(\backslashsqrt{x} = -8\) has no real number solution.

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Key Concepts

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

Square Root Function
The square root function, denoted as \(\sqrt{x}\), is a fundamental concept in mathematics. It represents a number that, when multiplied by itself, gives the original number \(x\). For example, \(\sqrt{16} = 4\) because \(4 \times 4 = 16\). The key property of the square root function is that it always yields a non-negative result. This means that \(\sqrt{x}\) can never be negative for any real number input. This is important because it restricts the possible values of the square root function to zero or positive numbers.
For instance:
  • \(\sqrt{9} = 3\)
  • \(\sqrt{0} = 0\)
    • These examples illustrate that no negative outputs are possible.
Non-Negative Values
Understanding non-negative values is crucial when dealing with the square root function. A non-negative value is a number that is either greater than or equal to zero. In simpler terms, non-negative values encompass all positive numbers and zero.
When we deal with the function \(\sqrt{x}\), we are limited to the range of non-negative values. This inherent property of the square root function is why the output can never be negative. For example:
  • \(\sqrt{25} = 5\), a positive number.
  • \(\sqrt{0} = 0\), a non-negative number.
  • There is no \(x\) such that \(\sqrt{x} = -1\) because -1 is not a non-negative value.
    • This restriction is why equations like \(\sqrt{x} = -8\) have no real solution.
Equation Analysis
When analyzing equations involving the square root function, the property's non-negative nature plays a pivotal role. Consider the equation \(\sqrt{x} = -8\). We can determine whether it has a real solution by looking at the properties of the square root function.
Knowing that \(\sqrt{x}\) produces only non-negative results, we see a contradiction in the equation \(\sqrt{x} = -8\). The left side, representing \(\sqrt{x}\), is inherently non-negative, whereas the right side is a negative value. This contradiction means there is no real number \(x\) that satisfies this equation.
To conclude:
  • The square root function cannot yield a negative result.
  • The equation \(\sqrt{x} = -8\) has no real solution because \(\sqrt{x}\) cannot equal a negative number.

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Most popular questions from this chapter

To estimate the speed at which a car was traveling at the time of an accident, a police officer drives the car under conditions similar to those during which the accident took place and then skids to a stop. If the car is driven at 30 mph, then the speed \(s\) at the time of the accident is given by $$ s=30 \sqrt{\frac{a}{p}} $$ where \(a\) is the length of the skid marks left at the time of the accident and \(p\) is the length of the skid marks in the police test. Find \(s\) for the following values of \(a\) and \(p .\) (a) \(a=862 \mathrm{ft} ; p=156 \mathrm{ft}\) (b) \(a=382 \mathrm{ft} ; p=96 \mathrm{ft}\) (c) \(a=84 \mathrm{ft} ; p=26 \mathrm{ft}\)

Simplify each radical. $$ \sqrt[3]{64 z^{6}} $$

Solve each equation. $$ x^{2}+4 x+3=0 $$

Simplify each radical. Assume that all variables represent nonnegative real numbers. $$ \sqrt{49 n^{2}} $$

Perform each operation and express the answer in simplest form. $$ (\sqrt[3]{5}-\sqrt[3]{4})(\sqrt[3]{25}+\sqrt[3]{20}+\sqrt[3]{16}) $$
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