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Chapter 3: Problem 10

For what positive values of \(x\) will \(x^{18}\) be equal to \(x^{20} ?\)

Short Answer

Expert verified
The positive value of \(x\) is 1.

Step by step solution

01

Set up the equation

Start by writing the equation given in the problem: \[ x^{18} = x^{20} \]
02

Simplify the equation

To make the equation easier to solve, divide both sides by \( x^{18} \): \[ \frac{x^{18}}{x^{18}} = \frac{x^{20}}{x^{18}} \] This simplifies to: \[ 1 = x^2 \]
03

Solve for \(x\)

To solve for \(x\), take the square root of both sides: \[ \sqrt{1} = \sqrt{x^2} \] This gives two possible solutions: \[ x = 1 \] and \[ x = -1 \]
04

Select the positive value

The problem states that we are looking for positive values of \(x\). Therefore, the correct solution is: \[ x = 1 \]

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Key Concepts

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

exponential functions
In this problem, solving such equations helps you understand how exponential growth or decay functions.
simplifying equations
We achieve a simpler equation that is much easier to work with. Simplifying in this manner takes practice but is essential for solving complex problems.
positive values
Understanding why we seek positive solutions helps provide a clearer perspective on problems and their implications.

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Most popular questions from this chapter

Challenge Consider \(\sqrt[n]{x}\) a. If \(\sqrt[n]{x}\) is positive, what can you say for sure about \(x ?\) Explain. b. If \(\sqrt[n]{x}\) is negative, what can you say for sure about \(n\) and \(x ?\) Explain.

Rewrite each expression as simply as you can. $$m^{-3} \cdot m^{4} \cdot b^{7}$$

Simplify each radical expression. If it is already simplified, say so. Challenge \(\sqrt{x+2}+\sqrt{4 x+8}\)

Rewrite each expression as simply as you can. $$4 a^{4} \cdot 3 a^{3}$$

Each table represents a linear relationship. Write an equation to represent each relationship. $$\begin{array}{|c|c|c|c|c|c|c|}\hline a & {-4} & {-3} & {-2} & {-1} & {0} & {1} \\ \hline b & {-8.8} & {-6.6} & {-4.4} & {-2.2} & {0} & {2.2} \\\ \hline\end{array}$$
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