Question

in exercises 27–34, solve the equation. check your solution(s). (see examples 5 and 6.)

1. $2x^{2/3} = 8$
2. $4x^{3/2} = 32$
3. $x^{1/4} + 3 = 0$
4. $2x^{3/4} - 14 = 40$
5. $(x + 6)^{1/2} = x$
6. $(5 - x)^{1/2} - 2x = 0$
7. $2(x + 11)^{1/2} = x + 3$
8. $(5x^2 - 4)^{1/4} = x$

error analysis in exercises 35 and 36, describe and correct the error in solving the equation.
35.
\boxed{\

$$\begin{aligned} \\sqrt3{3x - 8} &= 4 \\\\ (\\sqrt3{3x - 8})^3 &= 4 \\\\ 3x - 8 &= 4 \\\\ 3x &= 12 \\\\ x &= 4 \\end{aligned}$$

}
36.
\boxed{\

$$\begin{aligned} 8x^{3/2} &= 1000 \\\\ 8(x^{3/2})^{2/3} &= 1000^{2/3} \\\\ 8x &= 100 \\\\ x &= \\frac{25}{2} \\end{aligned}$$

}
ical functions

Explanation:

Problem 35: Error Analysis and Correction

The original solution incorrectly raised the left - hand side to the power of 3 but did not raise the right - hand side (4) to the power of 3. When we have an equation with a cube root, to eliminate the cube root, we should cube both sides of the equation.

1. Start with the equation \(\sqrt[3]{3x - 8}=4\).
2. Cube both sides of the equation: \((\sqrt[3]{3x - 8})^3=4^3\).

• By the property of cube roots, \((\sqrt[3]{a})^3 = a\), so the left - hand side simplifies to \(3x - 8\).
• The right - hand side, \(4^3=64\). So we get the equation \(3x - 8 = 64\).

1. Solve for \(x\):

• Add 8 to both sides: \(3x=64 + 8=72\).
• Divide both sides by 3: \(x=\frac{72}{3}=24\).

Problem 36: Error Analysis and Correction

The original solution made a mistake when raising both sides to the power of \(\frac{3}{2}\). The correct way is to first isolate \(x^{3/2}\) and then raise both sides to the power of \(\frac{2}{3}\) (the reciprocal of \(\frac{3}{2}\)) to solve for \(x\).

1. Start with the equation \(8x^{3/2}=1000\).
2. First, isolate \(x^{3/2}\) by dividing both sides by 8: \(x^{3/2}=\frac{1000}{8}=125\).
3. Now, raise both sides to the power of \(\frac{2}{3}\) to solve for \(x\). Recall that \((a^{m})^{n}=a^{mn}\). So \((x^{3/2})^{2/3}=x^{(3/2)\times(2/3)}=x\). And \(125^{2/3}=(5^{3})^{2/3}\).

• Using the property \((a^{m})^{n}=a^{mn}\), \((5^{3})^{2/3}=5^{3\times\frac{2}{3}} = 5^{2}=25\). So \(x = 25\).

Answer:

s:

• For problem 35: The error was not cubing the right - hand side. The correct solution gives \(x = 24\).
• For problem 36: The error was in the exponent operation when solving for \(x\). The correct solution gives \(x = 25\).