Question 1

Simplify the expression $$\sqrt[3]{128a$$^{15}$$}. $$Assume $$a \u003e 0.$$

Accepted Answer

$$4a^5$$

Question 2

Rewrite $$\sqrt[5]{y$$^{15}$$}$$ using rational exponents.

Accepted Answer

$$y^{3}$$

Question 3

Divide and simplify: $$\dfrac{6x^3$}{3x}$$, assuming $$x 
eq 0.$$

Accepted Answer

$$2x^2$$

Question 4

Multiply and simplify: $$(4 + 5x)(2 + x).$$

Accepted Answer

$$8 + 4x + 10x + 5x^2$ = 8 + 14x + 5x^2$

Question 5

Add and simplify: $$\sqrt{75} + 2\sqrt{12}.$$

Accepted Answer

$$5\sqrt{3} + 4\sqrt{3} = 9\sqrt{3}$$

Question 6

Rationalize and simplify: $$\dfrac{2}{\sqrt{5}}.$$

Accepted Answer

$$\dfrac{2\sqrt{5}}{5}$$

Question 7

Solve for $$x$$ and identify any extraneous solutions: $$\dfrac{x - 13}{x + 43} = 8.$$

Accepted Answer

The solution is $$x = 8x + 344 \implies x - 8x = 344 + 13 \implies -7x = 357 \implies x = -51$$; no extraneous solutions.

Question 8

Write the complex number $$(4 + i)(7 - i) in $$standard form $$a + bi.$$

Accepted Answer

$$29 + 3i$$

Question 9

Subtract and simplify: $$(4 + 9i) - (-3 - i).$$

Accepted Answer

$$7 + 10i$$

Question 10

Solve for $$x$$: $$64x^2 = 121$.

Accepted Answer

$$x = \pm\dfrac{11}{8}$$

Question 11

Rewrite the quadratic equation $$6x^2 + 19x = 20$ in standard form ax^2$ + bx + c = 0$$ and identify $$a$$, $$b$$, and $$c.$$

Accepted Answer

$$a = 6$$, $$b = 19$$, $$c = -20$$

Question 12

Solve using the quadratic formula: $$2x^2 - 9x - 1 = 0$. Give your answer in exact form.

Accepted Answer

$$x = \dfrac{9 \pm \sqrt{85}}{4}$$

Question 13

Multiply and simplify: $$\sqrt{27} \cdot \sqrt{12}.$$

Accepted Answer

$$18$$

Question 14

Rationalize and simplify: $$\dfrac{5}{2 - \sqrt{3}}.$$

Accepted Answer

$$\dfrac{10 + 5\sqrt{3}}{4 - 3}$$

Question 15

Solve for $$x$$ using the square root principle: $$(7x - 3)^2$ = 25.$$

Accepted Answer

$$x = \dfrac{3 \pm 5}{7}$$

Question 16

Solve for $$x$$: $$x^2 - x = -9$.

Accepted Answer

$$x = \dfrac{1 \pm \sqrt{37}}{2}$$

Question 17

Multiply and simplify: $$(4 + 9i)(-3 - i).$$

Accepted Answer

$$-12 - 4i - 27i - 9i^2$ = -12 - 31i + 9 = -3 - 31i$$

Question 18

Solve for $$x$$: $$8x^2 + 3 = -14x$.

Accepted Answer

$$x = \dfrac{14 \pm \sqrt{196 - 4 \cdot 8 \cdot 3}}{2 \cdot 8}$$

Question 19

Which of the following is equivalent to $$\sqrt[3]{189} in $$simplest radical form?

Accepted Answer

$$3\sqrt[3]{7}$$

Intermediate Algebra Practice Exam Guidance

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