(AA) Unit 2 Midterm (2024)

To simplify a complex fraction, multiply the numerator and the denominator by the least common denominator (LCD) of the fractions involved, then simplify the resulting expression.

Multiply the numerator and denominator by the conjugate of the denominator: $$\frac{(5+3i)(4+2i)}{(4-2i)(4+2i)} = \frac{20 + 10i + 12i - 6}{16 + 4} = \frac{14 + 22i}{20} = \frac{7 + 11i}{10}$$ .

The fourth root of a fraction $$\sqrt[4]{\frac{a}{b}}$$ is equal to $$\frac{\sqrt[4]{a}}{\sqrt[4]{b}}$$ .

Separate the fraction: $$\sqrt[4]{16} \cdot \sqrt[4]{a^4} \cdot \frac{1}{\sqrt[4]{c^3}} = 2 \cdot a \cdot \frac{1}{\sqrt[4]{c^3}} = \frac{2a}{\sqrt[4]{c^3}}$$ .

To factor a quadratic expression, look for two numbers that multiply to the constant term and add to the coefficient of the linear term.

Let $$u = 3x - 4$$ . Then the expression becomes $$u^2 + 9u + 18$$ . Factor to get $$(u + 3)(u + 6)$$ , substituting back gives $$(3x - 1)(3x + 2)$$ .

The conjugate is used to eliminate the imaginary part in the denominator, making it easier to simplify complex fractions.

Factor out $$x^{\frac{1}{4}}$$ : $$x^{\frac{1}{4}}(2x^{\frac{1}{4}} + 3) - 2 = 0$$ . This leads to the factors $$\left(2x^{\frac{1}{4}} - 1\right)(x^{\frac{1}{4}} + 2)$$ .

A complex number is a number that can be expressed in the form $$a + bi$$ , where $$a$$ and $$b$$ are real numbers and $$i$$ is the imaginary unit, defined as $$\sqrt{-1}$$ .

The difference of squares formula states that $$a^2 - b^2 = (a - b)(a + b)$$ .