How do you simplify the expression $$\frac{5+3i}{4-2i}$$ ?

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

How do you simplify $$\sqrt[4]{\frac{16a^4}{c^3}}$$ ?

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

How do you factor $$\left(3x-4\right)^2+9\left(3x-4\right)+18$$ ?

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

How do you factor $$2x^{\frac{1}{2}} + 3x^{\frac{1}{4}} - 2$$ ?

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

What is the definition of a complex number?

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 $$\sqrt{-1}$$ .

What is the formula for the difference of squares?

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

What is the square root of a variable raised to an exponent?

The square root of a variable raised to an exponent $$\sqrt{x^n}$$ is equal to $$x^{n/2}$$ .

How do you identify the roots of a quadratic equation?

The roots of a quadratic equation $$ax^2 + bx + c = 0$$ can be found using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ .

What is the process of rationalizing the denominator?

Rationalizing the denominator involves multiplying the numerator and denominator by a suitable expression to eliminate any radicals in the denominator.

What is the relationship between exponents and roots?

Exponents and roots are inversely related; for example, raising a number to a fractional exponent $$x^{\frac{m}{n}}$$ is equivalent to taking the nth root of $$x$$ raised to the mth power.

(AA) Unit 2 Midterm (2024) Flashcards

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What is the process of simplifying a complex fraction?

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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.

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