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Chapter 3: Problem 43

Find all real numbers \(x\) that satisfy the indicated equation. $$ x^{2 / 3}-6 x^{1 / 3}=-8 $$

Short Answer

Expert verified
The real values of x that satisfy the equation \(x^{\frac{2}{3}} - 6x^{\frac{1}{3}} = -8\) are \(x = 8\) and \(x = 64\).

Step by step solution

01

Make a substitution

Let's make a substitution to simplify the equation. We can set \(u = x^{\frac{1}{3}}\), so the equation becomes: \[u^2 - 6u = -8\]
02

Solve the Quadratic Equation

Now, let's solve the quadratic equation for u: \[u^2 - 6u + 8 = 0\] We need to factor this equation: \[(u - 2)(u - 4) = 0\] Now we got 2 possible values for u: \[u_1 = 2\] \[u_2 = 4\]
03

Reverse the Substitution and Find the Real Values of x

Now, let's reverse the substitution and find the corresponding values of x: Recalling that \(u = x^{\frac{1}{3}}\), we have: For \(u_1 = 2\): \[x^{\frac{1}{3}} = 2\] Cubing both sides, we get: \[x = 2^3 = 8\] For \(u_2 = 4\): \[x^{\frac{1}{3}} = 4\] Cubing both sides, we get: \[x = 4^3 = 64\] So, the real values of x that satisfy the given equation are: \[x = 8, 64\]

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Most popular questions from this chapter

Find the smallest integer \(n\) such that \(7^{n}>10^{100}\).

Suppose you have a calculator that can only compute square roots. Explain how you could use this calculator to compute \(7^{1 / 8}\).

Give an example of three irrational numbers \(x\), \(y,\) and \(z\) such that $$ \left(x^{y}\right)^{z} $$ is a rational number.

Find the smallest integer \(M\) such that \(5^{1 / M}<1.01\).

Using the result that \(\sqrt{2}\) is irrational (proved in Section 0.1), show that \(2^{5 / 2}\) is irrational.
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