Question 1

What is the least common denominator (LCD) for the rational expressions $$\frac{-7}{t}$$ and $$\frac{-7}{t+7}$$?

Accepted Answer

$$t(t+7)$$

Question 2

Find the LCD for $$\frac{1}{4a}$$ and $$\frac{-7}{a^2$+2a}.$$

Accepted Answer

Question 3

Determine the LCD for $$\frac{x-8}{x^2$+8x+12}$$ and $$\frac{x+3}{x^2$-3x-10}.$$

Accepted Answer

$$(x+6)(x+2)(x+5)(x-2)$$

Question 4

Simplify $$\frac{3}{13x} + \frac{1}{9x}$$ and choose the correct answer.

Accepted Answer

$$\frac{32}{117x}$$

Question 5

Simplify $$\frac{2}{r} + \frac{8}{r-9}$$ and choose the correct answer.

Accepted Answer

$$\frac{2(r-9)+8r}{r(r-9)}$$

Question 6

Simplify $$\frac{\frac{x}{2}}{\frac{7}{x+6}}.$$

Accepted Answer

$$\frac{x(x+6)}{14}$$

Question 7

Simplify $$\frac{9+\frac{3}{x}}{\frac{x}{4}+\frac{1}{12}}.$$

Accepted Answer

$$\frac{(9x+3)\cdot 12}{3x+1}$$

Question 8

Simplify $$\frac{\frac{25y^2$-9x^2$$}{xy}}{\frac{5}{x}-\frac{3}{y}}$.

Accepted Answer

$$\frac{(25y^2$-9x^2$$)}{xy(5y-3x)}$

Question 9

Simplify $$\frac{m$$^{-1}$$+z^{-1}$$}{$$m^{-1}-z$$^{-1}$$}.$$

Accepted Answer

$$\frac{m+z}{m-z}$$

Question 10

Simplify $$(a^{-1}+b$$^{-1}$$)-1$.

Accepted Answer

$$\frac{a+b-ab}{ab}$$

Question 11

A pharmacist needs to combine two solutions: one with concentration $$\frac{3}{13x}$$ and another with $$\frac{1}{9x}. $$What is the combined concentration?

Accepted Answer

$$\frac{32}{117x}$$

Question 12

A chemical reaction requires the sum of $$\frac{2}{r}$$ and $$\frac{8}{r-9}. $$What is the simplified form?

Accepted Answer

$$\frac{2(r-9)+8r}{r(r-9)}$$

Question 13

If $$x$$ represents the number of hours worked and $$x+6$$ the total hours including overtime, what is the simplified form of $$\frac{\frac{x}{2}}{\frac{7}{x+6}}$$?

Accepted Answer

$$\frac{x(x+6)}{14}$$

Question 14

A recipe requires $$9$$ cups of flour and $$\frac{3}{x}$$ cups of sugar. If the total is divided by $$\frac{x}{4}+\frac{1}{12}$$, what is the simplified form?

Accepted Answer

$$\frac{(9x+3)\cdot 12}{3x+1}$$

Question 15

A biologist is analyzing the ratio $$\frac{25y^2$-9x^2$$}{xy}$$ divided by $$\frac{5}{x}-\frac{3}{y}$. What is the simplified form?

Accepted Answer

$$\frac{(25y^2$-9x^2$$)}{xy(5y-3x)}$

Question 16

If $$m$$ and $$z$$ are variables representing two different populations, what is the simplified form of $$\frac{m$$^{-1}$$+z^{-1}$$}{$$m^{-1}-z$$^{-1}$$}$$?

Accepted Answer

$$\frac{m+z}{m-z}$$

Question 17

If $$a$$ and $$b$$ are rates of two machines, what is the simplified form of $$(a^{-1}+b$$^{-1}$$)-1$?

Accepted Answer

$$\frac{a+b-ab}{ab}$$

Intermediate Algebra Study Guide: Rational Expressions and Operations

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