MA104 Quiz - Chapters 3 and 4

What is the relationship between radicals and fractional exponents?

They are two spellings of the same idea: the nth root of a equals a^(1/n).

What does a^(m/n) mean?

The nth root of a, raised to the m: (n√a)^m. Example: 8^(2/3) = (cube-root-8)^2 = 2^2 = 4.

How do you simplify a radical like √80?

Pull out the largest perfect-square factor: √80 = √(16·5) = 4√5. ("Pull out everything that's allowed to leave.")

What is the product rule for radicals?

√a · √b = √(ab). Example: √3 · √12 = √36 = 6.

Why rationalize a denominator?

A radical in the denominator isn't wrong, it's just not the form everyone agreed to report answers in.

How do you rationalize 3/√6?

Multiply top and bottom by √6: 3√6/6 = √6/2.

How do you rationalize a binomial denominator like 2/(1+√3)?

Multiply top and bottom by the conjugate (1−√3): 2(1−√3)/(1−3) = 2(1−√3)/(−2) = √3−1.

What is a conjugate, and what does (a+b)(a−b) equal?

The same binomial with the opposite middle sign; (a+b)(a−b) = a² − b². The middle terms cancel, which kills the radical.

What are the steps for solving a radical equation?

1) Isolate the radical. 2) Raise both sides to the matching power. 3) Solve, then check every solution in the ORIGINAL equation.

What is an extraneous solution and where does it come from?

A "solution" that was never actually true, created when you raise both sides to a power. Reject it by checking against the original equation.

What does a^(−n) equal?

1/a^n. A negative exponent flips the base to the denominator.

Simplify 9^(3/2).

(√9)³ = 3³ = 27.

What is the one new twist when solving linear inequalities versus equations?

The inequality symbol flips direction when you multiply or divide both sides by a negative number.

Name four ways to report a solution set.

Inequality notation, set-builder notation, interval notation, and a graph.

Write −2 ≤ x ≤ 5 in interval notation.

[−2, 5]. Brackets mean the endpoints are included.

Write x > 3 in interval notation.

(3, ∞). Parenthesis means the endpoint is excluded; infinity always gets a parenthesis.

How do you solve a three-part "and" compound inequality?

Operate on all three parts at once. The solution is the overlap where both conditions are true.

How do you solve an "or" compound inequality?

Solve each part separately, then combine the solution sets with union (∪).

When do you use ∪ in interval notation?

For "or" compounds, when combining separate solution sets into one answer.

Solve: −2x > 6.

Divide by −2 and flip the symbol: x < −3.

What exactly is on the Wed 10/7 MA104 quiz?

Chapter 3 (3.1–3.4: radicals and rational exponents) and Chapter 4 (4.1–4.2: linear and compound inequalities). Section 4.3 absolute value is NOT on it.

When is the quiz given, and what is still unknown about it?

In class Wednesday 10/7, at the beginning of class (per the 9/30 precedent). The number of questions and the calculator policy were not posted.