H13B Exam 1 Review

Integral exponents

100

Simplify: x⁹ · x^-4

x⁵

200

Evaluate: 3^-2

1/9  (not -9)

300

Simplify with positive exponents only:

(2x^-3y⁵) / (8x²y^-1)

y⁶ / (4x⁵)

400

Simplify with positive exponents only:

(3a^-2b⁴)^-3

a⁶ / (27b¹²)

500

Simplify with positive exponents only:

( (6x⁵y^-2z) / (2x^-3y⁴z^-11) )^-3

y¹⁸ / (27x²⁴z³⁶)

Simplify inside first: 3x⁸y^-6z¹²

GCF & grouping

100

Factor: 14x⁴ - 21x³

7x³(2x - 3)

200

Factor: 9a³b² - 15a²b³ + 6a²b²

3a²b²(3a - 5b + 2)

300

Factor by grouping:

x³ - 2x² + 5x - 10

(x - 2)(x² + 5)

400

Factor by grouping:

12xy - 10y - 18x + 15

(6x - 5)(2y - 3)

The minus pulled out of the second pair is what makes (6x - 5) appear.

500

Factor completely:

x³ + 5x² - 16x - 80

(x + 5)(x - 4)(x + 4)

Grouping gives (x + 5)(x² - 16) — and that factors again.

Trinomials & substitution

100

Factor: x² + 7x + 12

(x + 3)(x + 4)

200

Factor: x² - 2x - 35

(x - 7)(x + 5)

300

Factor: 6x² + x - 12

(3x - 4)(2x + 3)

Middle check: 9x - 8x = x

400

Factor: 8x² - 2x - 15

(4x + 5)(2x - 3)

Middle check: -12x + 10x = -2x

500

Factor completely:

4x³ - 14x² - 30x

2x(2x + 3)(x - 5)

GCF of 2x first. Without it, 2x² - 7x - 15 is hiding.

Special binomials

100

Factor: x² - 81

(x - 9)(x + 9)

200

Factor: 49x² - 100y²

(7x - 10y)(7x + 10y)

300

Factor: x³ + 64

(x + 4)(x² - 4x + 16)

Sum of cubes. Signs: same, opposite, always plus.

400

Factor completely:

x³y - 8y

y(x - 2)(x² + 2x + 4)

GCF of y first, then a difference of cubes.

500

Factor completely:

8x⁶ + 27y³

(2x² + 3y)(4x⁴ - 6x²y + 9y²)

8x⁶ = (2x²)³ and 27y³ = (3y)³

Rational expressions

100

Find the domain:

(x + 3) / (x - 5)

All real numbers except x = 5

{ x | x ≠ 5 }

200

Simplify and state the restrictions:

(x² - 25) / (x² + 9x + 20)

(x - 5)/(x + 4),  x ≠ -4, -5

Restrictions come from the original denominator — including the factor that cancelled.

300

Multiply and simplify:

(x² - 9)/(2x + 8) · (x + 4)/(x - 3)

(x + 3)/2

No common denominator needed — multiplication never needs one.

400

Subtract and simplify:

(4x - 4)/(x² + 2x - 15) - 3/(x + 5)

1/(x - 3)

Factor the bottom to (x + 5)(x - 3) first — the LCD is already there. Top: 4x - 4 - 3(x - 3) = x + 5.

500

Simplify the complex rational expression:

(1/x - 1/y) / (1 - x²/y²)

y / (x(y + x))

Top becomes (y - x)/(xy), bottom becomes (y - x)(y + x)/y². Flip and multiply — the (y - x) cancels.