Operations with Radicals and Exponents

Laws of Exponents

Rule	Formula	Example
Product	$a^m \cdot a^n = a^{m+n}$	$x^3 \cdot x^5 = x^8$
Quotient	$\frac{a^m}{a^n} = a^{m-n}$	$\frac{x^7}{x^2} = x^5$
Power of a power	$(a^m)^n = a^{mn}$	$(x^3)^4 = x^{12}$
Power of a product	$(ab)^n = a^n b^n$	$(2x)^3 = 8x^3$
Power of a quotient	$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$	$\left(\frac{x}{3}\right)^2 = \frac{x^2}{9}$
Zero exponent	$a^0 = 1$ (if $a \neq 0$)	$5^0 = 1$
Negative exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-3} = \frac{1}{x^3}$

Rational Exponents

The connection between exponents and radicals:

$a^{\frac{1}{n}} = \sqrt[n]{a}$

$a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Examples:

• $x^{\frac{1}{2}} = \sqrt{x}$
• $x^{\frac{2}{3}} = \sqrt[3]{x^2}$
• $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$

Simplifying Radicals

$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$

Example: $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$

Example: $\sqrt{50x^4} = \sqrt{25 \cdot 2 \cdot x^4} = 5x^2\sqrt{2}$

Rationalising the Denominator

To remove a radical from the denominator, multiply by a form of 1:

$\frac{5}{\sqrt{3}} = \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$

For binomial denominators: multiply by the conjugate.

$\frac{1}{\sqrt{5} + 2} \cdot \frac{\sqrt{5} - 2}{\sqrt{5} - 2} = \frac{\sqrt{5} - 2}{5 - 4} = \sqrt{5} - 2$

Adding and Subtracting Radicals

Combine like radicals (same radicand):

$3\sqrt{5} + 7\sqrt{5} = 10\sqrt{5}$

$\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$

Multiplying Radicals

$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$

$3\sqrt{2} \cdot 4\sqrt{6} = 12\sqrt{12} = 12 \cdot 2\sqrt{3} = 24\sqrt{3}$

Tip 1: Memorise the Laws of Exponents

These appear on virtually every SAT. Know them cold.

Tip 2: Convert Radicals to Exponents When Needed

$\sqrt{x}$ and $x^{1/2}$ are the same. Sometimes exponent form is easier to manipulate.

Tip 3: Factor Out Perfect Squares

When simplifying $\sqrt{n}$, find the largest perfect square factor of $n$.

Tip 4: Don't Mix Up Exponent Rules

$x^3 \cdot x^5 = x^8$ (add exponents). $(x^3)^5 = x^{15}$ (multiply exponents). Don't confuse these.

Tip 5: Negative Exponents = Reciprocal

$2^{-3} = \frac{1}{8}$, not $-8$. The negative means reciprocal, not negative value.