2.6 Other Types of Equations

Evaluate $8^{\frac{2}{3}} .$

Evaluate $64^{- \frac{1}{3}} .$

Solve the equation in which a variable is raised to a rational exponent: $x^{\frac{5}{4}} = 32.$

Solve the equation $x^{\frac{3}{2}} = 125.$

Solve $3 x^{\frac{3}{4}} = x^{\frac{1}{2}} .$

Solve: $\left(x + 5\right)^{\frac{3}{2}} = 8.$

Solve the polynomial by factoring: $5 x^{4} = 80 x^{2} .$

Solve by factoring: $12 x^{4} = 3 x^{2} .$

Solve $\sqrt{15 - 2 x} = x .$

Solve the radical equation: $\sqrt{x + 3} = 3 x - 1$

Solve $\sqrt{2 x + 3} + \sqrt{x - 2} = 4.$

Solve the equation with two radicals: $\sqrt{3 x + 7} + \sqrt{x + 2} = 1.$

Solve the following absolute value equations:

1. ⓐ $\left|6 x + 4\right| = 8$
2. ⓑ $\left|3 x + 4\right| = −9$
3. ⓒ $\left|3 x - 5\right| - 4 = 6$
4. ⓓ $\left|−5 x + 10\right| = 0$

Solve the absolute value equation: $\left|1 - 4 x\right| + 8 = 13.$

Solve this fourth-degree equation: $3 x^{4} - 2 x^{2} - 1 = 0.$

Solve using substitution: $x^{4} - 8 x^{2} - 9 = 0.$

Solve the equation in quadratic form: $\left(x + 2\right)^{2} + 11 \left(x + 2\right) - 12 = 0.$

Solve: $\left(x - 5\right)^{2} - 4 \left(x - 5\right) - 21 = 0.$

Solve the following rational equation: $\frac{- 4 x}{x - 1} + \frac{4}{x + 1} = \frac{- 8}{x^{2} - 1} .$

Solve $\frac{3 x + 2}{x - 2} + \frac{1}{x} = \frac{- 2}{x^{2} - 2 x} .$

In a radical equation, what does it mean if a number is an extraneous solution?

Your friend tries to calculate the value $- 9^{\frac{3}{2}}$ and keeps getting an ERROR message. What mistake are they probably making?

Explain how to change a rational exponent into the correct radical expression.

Solve the rational exponent equation. Use factoring where necessary.

$x^{\frac{3}{4}} = 27$

Solve the rational exponent equation. Use factoring where necessary.

$\left(x - 1\right)^{\frac{3}{4}} = 8$

Solve the rational exponent equation. Use factoring where necessary.

$x^{\frac{2}{3}} - 5 x^{\frac{1}{3}} + 6 = 0$

Solve the following polynomial equations by grouping and factoring.

$x^{3} + 2 x^{2} - x - 2 = 0$

Solve the following polynomial equations by grouping and factoring.

$4 y^{3} - 9 y = 0$

Solve the following polynomial equations by grouping and factoring.

$m^{3} + m^{2} - m - 1 = 0$

Solve the following polynomial equations by grouping and factoring.

$5 x^{3} + 45 x = 2 x^{2} + 18$

Solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.

$\sqrt{x - 7} = 5$

Solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.

$\sqrt{3 t + 5} = 7$

Solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.

$\sqrt{12 - x} = x$

Solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.

$\sqrt{3 x + 7} + \sqrt{x + 2} = 1$

Solve the equation involving absolute value.

$\left|3 x - 4\right| = 8$

Solve the equation involving absolute value.

$\left|1 - 4 x\right| - 1 = 5$

Solve the equation involving absolute value.

$\left|2 x - 1\right| - 7 = −2$

Solve the equation involving absolute value.

$\left|x + 5\right| = 0$

Solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.

$x^{4} - 10 x^{2} + 9 = 0$

Solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.

$\left(x^{2} - 1\right)^{2} + \left(x^{2} - 1\right) - 12 = 0$

Solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.

$\left(x - 3\right)^{2} - 4 = 0$

Solve for the unknown variable.

$\sqrt{\left|x\right|^{2}} = x$

Solve for the unknown variable.

$\left|x^{2} + 2 x - 36\right| = 12$

Use the model for the period of a pendulum, $T ,$ such that $T = 2 \pi \sqrt{\frac{L}{g}} ,$ where the length of the pendulum is L and the acceleration due to gravity is $g .$

If the gravity is 32 ft/s² and the period equals 1 s, find the length to the nearest in. (12 in. = 1 ft). Round your answer to the nearest in.

Use a model for body surface area, BSA, such that $B S A = \sqrt{\frac{w h}{3600}} ,$ where w = weight in kg and h = height in cm.

Find the weight of a 177-cm male to the nearest kg whose $B S A = 2.1.$