Exponents and Radicals

Flashcards covering definitions, exponent rules, properties of roots, simplifying radical expressions, rational exponents, and standard form rationalization.

In exponential notation, what is defined as the exponent?

The exponent of a term is the power that the term is being raised to.

In exponential notation, what is defined as the base?

The base is the term that is being raised to an exponent.

What is the Product Rule for exponents?

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

What is the Power Rule for exponents?

$$(a^n)^m = a^{n \cdot m}$$

How are negative exponents defined for a real number $a$ and positive integer $n$?

$$a^{-n} = \frac{1}{a^n}$$

What is the Quotient Rule for exponents?

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

What is the Product Distribution rule for exponents?

$$(ab)^n = a^n b^n$$

What is the Quotient Distribution rule for exponents?

$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

What is the value of $a^0$ for any real number $a$, and why is $0^0$ undefined?

$a^0 = 1$ for any non-zero real number $a$. $0^0$ is undefined because attempting the rule $0^0 = 0^{1-1} = \frac{0^1}{0^1} = \frac{0}{0}$ involves dividing by $0$, which is not allowed.

What is the formal definition of a number $x$ written in scientific notation?

A number $x$ is written in scientific notation if it is in the form $x = a \times 10^n$ where $1 \le a < 10$ and $n \in \mathbb{Z}$.

Based on the given figure, what algebraic equation represents the side length $x$ of the square?

$$x^2 = 11$$

What is the formal definition of a square root of a number $b$?

For numbers $a$ and $b$, $a$ is a square root of $b$ if $a^2 = b$.

What is the definition of an $n\text{th}$ root of a number $b$?

For numbers $a$ and $b$, $a$ is an $n\text{th}$ root of $b$ if $a^n = b$. The value $n$ is called the index of the root.

What is the principal square root of a real number $a$?

If a real number $a$ has a square root, its principal square root is the one that has the same sign as $a$, written as $\sqrt{a}$.

What is the principal $n\text{th}$ root of a real number $a$?

If a real number $a$ has an $n\text{th}$ root, its principal $n\text{th}$ root is the one that has the same sign as $a$, written as $\sqrt[n]{a}$. Here, $a$ is the radicand and $n$ is the index.

What is the general rule for simplifying $\sqrt[n]{a^n}$ depending on whether $n$ is odd or even?

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

What is the Product Rule for roots?

For positive numbers $a, b$ and positive integer $n$, $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$.

What is the Quotient Rule for roots?

For positive numbers $a, b$ and positive integer $n$, $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$.

What is the rule for nested roots?

For positive number $a$ and positive integers $m, n$, $\sqrt[m]{\sqrt[n]{a}} = \sqrt[m \cdot n]{a}$.

When can radical expressions be simplified via addition or subtraction?

Radicals cannot be simplified via addition or subtraction unless the radicands and indices are the same, following $a\sqrt[n]{x} + b\sqrt[n]{x} = (a + b)\sqrt[n]{x}$.

What three conditions must be met for a radical expression to be simplified?

(a) All possible factors have been taken out, (b) denominators are radical-free, and (c) the index of the radical is reduced.

How are rational exponents defined in terms of radical notation?

For any natural number $n$, natural number $m$, and real number $a$, $a^{1/n} = \sqrt[n]{a}$ and $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$.

What is standard form for a fraction involving radicals?

A fraction whose denominator contains no radicals is said to be in standard form.

How do you rationalize a denominator of the form $\sqrt[n]{a^m} = a^{m/n}$ where $a > 0$?

Multiply both the numerator and denominator by $\sqrt[n]{a^{n-m}} = a^{(n-m)/n}$, so that the exponent in the denominator sums to $\frac{m}{n} + \frac{n-m}{n} = \frac{n}{n} = 1$.