Notes
Exponent rules
- Same base: $a^m \cdot a^n = a^{m+n}$, $\displaystyle \frac{a^m}{a^n} = a^{m-n}$, $(a^m)^n = a^{mn}$.
- $a^0 = 1$ and $\displaystyle a^{-n} = \frac{1}{a^n}$ for $a \neq 0$.
- Apply a power to every factor inside parentheses: $(3x^2y)^3 = 27x^6y^3$.
Rational exponents and radicals
- $\displaystyle a^{m/n} = \left(\sqrt[n]{a}\right)^m$. The denominator is the root and the numerator is the power.
- Simplify a radical by pulling out perfect squares (or cubes): $\displaystyle \sqrt{72} = 6\sqrt{2}$.
- Add or subtract only like radicals. Multiply with $\displaystyle \sqrt{a}\sqrt{b} = \sqrt{ab}$. Rationalize denominators by multiplying by the radical.
Rational vs irrational
- Rational numbers are fractions of integers, with decimals that end or repeat. Irrational numbers like $\displaystyle \sqrt{2}$ and $\pi$ do not.
- Rational + irrational is irrational. Nonzero rational $\times$ irrational is irrational. Two irrationals can multiply to a rational.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Cube each factor: $3^3 = 27$, $(x^2)^3 = x^6$, $y^3$.
Subtract exponents of $x$: $-3 - (-5) = 2$. So the result is $x^2y^2$.
Cube root of 27 is 3, then $3^2 = 9$.
$98 = 49 \cdot 2$ and $\displaystyle \sqrt{49} = 7$.
$\displaystyle \sqrt{5}\cdot\sqrt{5} = 5$ and $\displaystyle \sqrt{5}\cdot\sqrt{20} = \sqrt{100} = 10$. The total is $15$.
$\displaystyle 0.\overline{3} = \frac{1}{3}$, $\displaystyle \sqrt{81} = 9$, and $\displaystyle -\frac{7}{3}$ is a fraction. 12 is not a perfect square, so $\displaystyle \sqrt{12}$ is irrational.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A square garden has an area of 200 square meters.
(a) Find the side length in simplest radical form.
Show the answer and scoring guide
Side $\displaystyle = \sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}$ meters.
Check off the points you earned:
(b) Find the perimeter in exact form and as a decimal rounded to the nearest hundredth.
Show the answer and scoring guide
Perimeter $\displaystyle = 4 \cdot 10\sqrt{2} = 40\sqrt{2} \approx 56.57$ meters.
Check off the points you earned:
(c) Is the side length rational or irrational? Explain.
Show the answer and scoring guide
It is irrational. 200 is not a perfect square, so $\displaystyle 10\sqrt{2}$ is a nonzero rational number times the irrational $\displaystyle \sqrt{2}$, which is irrational.
Check off the points you earned: