Algebraic Tips

Question 1

If ab = 0, what does this tell you?

Answer 1

a = 0 or b = 0 (or both)

Question 2

When should you substitute a variable (like t = tan(x))?

Answer 2

When equation has mix of tan, cot, sec — convert everything to one variable to simplify algebra

Question 3

What is any number divided by zero equal to?

Answer 3

Undefined - you cannot divide by zero

Question 4

How do you evaluate (√2)^10

Answer 4

It’s 10 because it’s 5 pairs of 2

Question 5

What’s the rule for √a x √b

Answer 5

It’s equal to √(ab)
a and b >= 0 or it doesn’t work

Question 6

“When dividing both sides of an equation by a variable (e.g., dividing by x), what must you be certain of to ensure the step is mathematically valid?

Answer 6

You must be certain that the variable does not equal zero. If not, factor instead

Question 7

What does a ≡ b (mod n) mean?

Answer 7

“a is congruent to b modulo n” - means a and b have the same remainder when divided by n.

Question 8

What does n | (a - b) mean in relation to mod?

Answer 8

a ≡ b (mod n) if and only if n divides (a - b). They’re equivalent statements.

Question 9

What are the possible values of n (mod 3)?

Answer 9

n ≡ 0 (mod 3), n ≡ 1 (mod 3), or n ≡ 2 (mod 3)
(These are the only possible remainders when dividing by 3)

Question 10

How do you write “n is divisible by 3” using mod notation?

Answer 10

How do you write “n is divisible by 3” using mod notation?

Question 11

Rule - If mn is even, then what?

Answer 11

At least one of m or n must be even. (Contrapositive: if both m and n are odd, then mn is odd)

Question 12

Rule - Three consecutive integers:

Answer 12

Among any 3 consecutive integers, at least one is divisible by 3. (General idea: consecutive integers cycle through remainders mod 3)

Question 13

How do you convert scientific notation to expanded form?

Answer 13

Move the decimal point based on the exponent. Positive exponent → move right (making number bigger). Negative exponent → move left (making number smaller). Add zeros as needed.

Question 14

What is (n+1)! = ?

Answer 14

n! + (n+1)