Notes
Roots and rational exponents
- $\displaystyle \sqrt[n]{a}$ is the number whose $n$th power is $a$. Odd roots of negatives are real; even roots of negatives are not.
- $\displaystyle a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$: denominator is the root, numerator is the power.
- $\displaystyle a^{-r} = \frac{1}{a^r}$, and all exponent rules still apply.
Simplifying radicals
- Pull out perfect powers: $\displaystyle \sqrt{48} = 4\sqrt{3}$, $\displaystyle \sqrt[3]{40} = 2\sqrt[3]{5}$.
- Combine only like radicals: $\displaystyle 2\sqrt{5} + 7\sqrt{5} = 9\sqrt{5}$.
- Rationalize denominators; use the conjugate for binomials: $\displaystyle (a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$.
Solving radical equations
1. Isolate the radical. 2. Raise both sides to the index power. 3. Solve, then check every answer to throw out extraneous solutions.
Graphs
| Function | Domain | Range | Key point |
|---|---|---|---|
| $\displaystyle y = \sqrt{x - h} + k$ | $x \ge h$ | $y \ge k$ | starts at $(h, k)$ |
| $\displaystyle y = \sqrt[3]{x - h} + k$ | all reals | all reals | center $(h, k)$ |
Topics in this unit
Each topic has its own short notes and practice.
Practice
$\displaystyle 27^{\frac{2}{3}} = 3^2 = 9$, and the negative exponent gives $\displaystyle \frac{1}{9}$.
$16x^4 = 8x^3 \cdot 2x$, and $\displaystyle \sqrt[3]{8x^3} = 2x$.
Conjugates: $2^2 - 3 = 1$.
$\displaystyle \sqrt{x - 2} = -3$, but a square root can't be negative. Squaring gives $x = 11$, which fails the check.
$6 - 2x \ge 0 \Rightarrow -2x \ge -6 \Rightarrow x \le 3$. Flip the sign when dividing by a negative.
The root index 5 goes in the denominator.
Free response
Write your answer first, then open the scoring guide.
Free-response question
When an object is dropped from a height of $h$ feet, the time in seconds it takes to hit the ground is about $t = \sqrt{\frac{h}{16}}$.
(a) How long does it take an object dropped from 144 feet to land?
Show the answer and scoring guide
$\displaystyle t = \sqrt{\frac{144}{16}} = \sqrt{9} = 3$ seconds.
Check off the points you earned:
(b) A stone takes 2.5 seconds to land. From what height was it dropped?
Show the answer and scoring guide
$\displaystyle 2.5 = \sqrt{\frac{h}{16}} \Rightarrow 6.25 = \frac{h}{16} \Rightarrow h = 100$ feet.
Check off the points you earned:
(c) How high must an object be dropped to take twice as long as in part (a)? How does this height compare with 144 feet?
Show the answer and scoring guide
$\displaystyle 6 = \sqrt{\frac{h}{16}} \Rightarrow 36 = \frac{h}{16} \Rightarrow h = 576$ feet. This is $\displaystyle \frac{576}{144} = 4$ times as high: doubling the time takes 4 times the height.
Check off the points you earned: