Q: Simplify the rational expression and state the domain: $$\dfrac{3x - 2}{15x^2$ + 22x + 8}$$

$$\dfrac{3x - 2}{(3x + 4)(5x + 2)},\ x \neq -\dfrac{4}{3},\ x \neq -\dfrac{2}{5}$$

Q: Given the formula $$N = 4x^2$ + 5x + 3$$ for the number of households $$N (in $$thousands) with computers $$x$$ years after 1990, in what year were there 87 thousand households with computers?

$$x = 4$$,\ \text{so year is } 1994$

Question 1

Simplify the exponential expression: $$(-5x^3y$$^{-4}$$)(2x^{-1}y$$)$$. Assume that variables represent nonzero real numbers.

Accepted Answer

$$-10x^2y$$^{-3}$

Question 2

Simplify: $$\dfrac{2$$^{-8}$$x^{-1}4$$}{$$2^{-5}x$$^{-4}$$8}$$

Accepted Answer

$$x^3$$

Question 3

Simplify: $$\left(\dfrac{x^5$y^0$$}{$$x^6y$$}\$$right)^{-2}$

Accepted Answer

$$\dfrac{y^2$}{x^2$}$$

Question 4

Simplify: $$\dfrac{17.08 	imes 10$$^{-8}$$}{4 	imes 10^5$}$$ and write the answer in scientific notation.

Accepted Answer

$$4.27 	imes 10$$^{-13}$$

Question 5

Add or subtract: $$\sqrt{3x} + 7\sqrt{48x} - 3\sqrt{108x}$$

Accepted Answer

$$15\sqrt{3x}$$

Question 6

Rationalize the denominator: $$\dfrac{\sqrt{6}}{\sqrt{17} + 2}$$

Accepted Answer

$$\dfrac{\sqrt{102} - 2\sqrt{6}}{13}$$

Question 7

Simplify using properties of exponents: $$(6x^{1/3})(3x$$^{3/2}$$)$

Accepted Answer

$$18x^{11/6}$$

Question 8

Factor completely: $$4x^2 - 49$

Accepted Answer

$$(2x + 7)(2x - 7)$$

Question 9

Factor completely: $$27x^3 + 64$

Accepted Answer

$$(3x + 4)(9x^2$ - 12x + 16)$$

Question 10

Simplify the rational expression and state the domain: $$\dfrac{3x - 2}{15x^2$ + 22x + 8}$$

Accepted Answer

$$\dfrac{3x - 2}{(3x + 4)(5x + 2)},\ x 
eq -\dfrac{4}{3},\ x 
eq -\dfrac{2}{5}$$

Question 11

Solve for $$x$$: $$\dfrac{x - 6}{3x} = \dfrac{x + 5}{x} - 8 ($$State any restrictions on $$x.)$$

Accepted Answer

$$x = -1,\ x 
eq 0$$

Question 12

Solve the equation: $$9x/8 - x + x/40 = 4/5$$

Accepted Answer

$$x = 4$$

Question 13

Solve the absolute value inequality: $$|x - 5| - 7 \leq 14$$

Accepted Answer

$$[-2, 12]$$

Question 14

Solve the radical equation and check all proposed solutions: $$\sqrt{8x + 8} = x + 4$$

Accepted Answer

$$\{8\}$$

Question 15

Solve the linear inequality and express the solution in interval notation: $$-14x + 12 \u003c -2(6x - 10)$$

Accepted Answer

$$[-4, \infty)$$

Question 16

Solve the absolute value inequality: $$5 + |1 - \dfrac{x}{2}| \geq 8$$

Accepted Answer

$$(-\infty, -4] \cup [8, \infty)$$

Question 17

Given the formula $$N = 4x^2$ + 5x + 3$$ for the number of households $$N (in $$thousands) with computers $$x$$ years after 1990, in what year were there 87 thousand households with computers?

Accepted Answer

$$x = 4$$,\ 	ext{so year is } 1994$

Precalculus Review: Exponents, Radicals, Factoring, Rational Expressions, and Equations

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