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Chapter 1: Problem 32

Solve and check each equation with rational exponents. $$ (x+5)^{3 / 2}=8 $$

Short Answer

Expert verified
The solution to the equation is \( x = -1 \)

Step by step solution

01

Isolate the term with the rational exponent

Rewrite the equation by isolating the term with the rational exponent on one side: \[ (x+5)^{3 / 2} = 8\]
02

Apply the square root to both sides of the equation

We can take the 2/3 power on both sides to eliminate the 3/2 power on the left hand side. That will result in a linear equation: \[ (x+5) = \sqrt[3]{8^2}=\sqrt[3]{64} \]
03

Solve the Linear Equation

The cube root of 64 is 4, our equation now becomes \( x + 5 = 4 \). Solve this algebraic linear equation to find the solution \(x = 4 - 5 = -1\)
04

Check the solution

To validate the solution, we substitute \(x\) with \(-1\) in the original equation: \[ ((-1) + 5)^{3 / 2} = 8.\] Simplifying this results to \(8 = 8\) indicating that our solution is indeed correct

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Most popular questions from this chapter

Explain how to add complex numbers. Provide an example with your explanation.

Which one of the following is true? a. Some irrational numbers are not complex numbers. b. \((3+7 i)(3-7 i)\) is an imaginary number. c. \(\frac{7+3 i}{5+3 i}=\frac{7}{5}\) d. In the complex number system, \(x^{2}+y^{2}\) (the sum of two squares) can be factored as \((x+y i)(x-y i)\)

Write a quadratic inequality whose solution set is \([-3,5]\)

In Exercises \(1-8,\) add or subtract as indicated and write the result in standard form. $$6-(-5+4 i)-(-13-11 i)$$

In Exercises \(9-20,\) find each product and write the result in standard form. $$(-4-8 i)(3+9 i)$$
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