Lesson 6: Chapter Summary and Review

New Concept

This lesson unifies exponents and radicals. You'll see how fractional exponents, like $x^{1/2}$, are just another way to write roots, like $\sqrt{x}$. We'll use this connection to simplify expressions and solve new types of equations.

What’s next

Next, you'll tackle interactive examples and practice problems covering exponent laws, radical simplification, and solving equations with roots and powers.

Property

Definition of Negative and Zero Exponents.

$a^{-n} = \frac{1}{a^n}$ $(a \neq 0)$

$a^0 = 1$ $(a \neq 0)$

Property

Laws of Exponents.

I. $a^m \cdot a^n = a^{m+n}$

II. $\frac{a^m}{a^n} = a^{m-n}$

Property

To Write a Number in Scientific Notation.

1. Locate the decimal point so that there is exactly one nonzero digit to its left.

2. Count the number of places you moved the decimal point; this determines the power of 10.

a. If the original number is greater than 10, the exponent is positive.
b. If the original number is less than 1, the exponent is negative.

Property

Exponential Notation for Radicals.
For any integer $n \geq 2$ and for $a \geq 0$,
$a^{1/n} = \sqrt[n]{a}$

Rational Exponents.
$a^{m/n} = (a^{1/n})^m = (a^m)^{1/n}$, $a > 0$, $n \neq 0$

Rational Exponents and Radicals.
$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Property

Product Rule for Radicals.
$\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}$, for $a, b \geq 0$

Quotient Rule for Radicals.
$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$, for $a \geq 0$, $b > 0$

In general, it is not true that $\sqrt{a + b}$ is equivalent to $\sqrt{a} + \sqrt{b}$.

Property

Roots of Powers.

$\sqrt[n]{a^n} = a$ if $n$ is odd.

$\sqrt[n]{a^n} = |a|$ if $n$ is even.