Exponents and Radicals (SAT Math)

Exponent Rules

Rule 1: Product Rule

When multiplying same base, ADD exponents:

$x^a \cdot x^b = x^{a+b}$

Examples:

• $x^3 \cdot x^5 = x^{3+5} = x^8$
• $2^4 \cdot 2^3 = 2^{4+3} = 2^7 = 128$

Rule 2: Quotient Rule

When dividing same base, SUBTRACT exponents:

$\frac{x^a}{x^b} = x^{a-b}$

Examples:

• $\frac{x^7}{x^4} = x^{7-4} = x^3$
• $\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$

Rule 3: Power Rule

When raising power to power, MULTIPLY exponents:

$(x^a)^b = x^{ab}$

Examples:

• $(x^3)^4 = x^{3 \cdot 4} = x^{12}$
• $(2^2)^3 = 2^{2 \cdot 3} = 2^6 = 64$

Rule 4: Power of a Product

Distribute the exponent:

$(xy)^a = x^a y^a$

Examples:

• $(3x)^2 = 3^2 \cdot x^2 = 9x^2$
• $(2ab)^3 = 2^3 a^3 b^3 = 8a^3b^3$

Rule 5: Power of a Quotient

Distribute the exponent:

$\left(\frac{x}{y}\right)^a = \frac{x^a}{y^a}$

Examples:

• $\left(\frac{x}{3}\right)^2 = \frac{x^2}{3^2} = \frac{x^2}{9}$
• $\left(\frac{2}{y}\right)^4 = \frac{2^4}{y^4} = \frac{16}{y^4}$

Special Exponents

Zero Exponent

Any number (except 0) to the zero power equals 1:

$x^0 = 1$ (where $x \neq 0$)

Examples:

• $5^0 = 1$
• $(x^2y^3)^0 = 1$
• $-2(3x)^0 = -2(1) = -2$ (only $(3x)^0 = 1$)

Negative Exponents

Negative exponent means reciprocal:

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

Examples:

• $x^{-3} = \frac{1}{x^3}$
• $2^{-4} = \frac{1}{2^4} = \frac{1}{16}$
• $\frac{1}{x^{-2}} = x^2$ (flipping reciprocal)

In fractions: $\frac{x^{-2}}{y^{-3}} = \frac{y^3}{x^2}$

Negative exponents "flip" to the other part of the fraction

Fractional Exponents

Fractional exponent = root:

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

Examples:

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

General form: $x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$

Examples:

• $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$
• $16^{\frac{3}{4}} = (\sqrt[4]{16})^3 = 2^3 = 8$

Radical Rules

Simplifying Radicals

Product Rule: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$

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

Quotient Rule: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

Example: $\sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}$

Simplification Strategy

Find perfect square factors:

Example: Simplify $\sqrt{48}$

Step 1: Factor into perfect square
$48 = 16 \cdot 3$

Step 2: Split the radical
$\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3}$

Step 3: Simplify
$= 4\sqrt{3}$

Common perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

Adding and Subtracting Radicals

Can only combine LIKE radicals (same radicand):

✓ $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$
✓ $7\sqrt{3} - 2\sqrt{3} = 5\sqrt{3}$
❌ $\sqrt{2} + \sqrt{3}$ cannot be simplified (different radicands)

Sometimes need to simplify first:

Example: $\sqrt{50} + \sqrt{8}$

$\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$ $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$

$\sqrt{50} + \sqrt{8} = 5\sqrt{2} + 2\sqrt{2} = 7\sqrt{2}$

Multiplying Radicals

Multiply coefficients and radicands separately:

$a\sqrt{b} \cdot c\sqrt{d} = (a \cdot c)\sqrt{b \cdot d}$

Examples:

• $2\sqrt{3} \cdot 5\sqrt{2} = 10\sqrt{6}$
• $3\sqrt{6} \cdot 2\sqrt{6} = 6\sqrt{36} = 6(6) = 36$

Rationalizing the Denominator

Don't leave radicals in denominator:

$\frac{1}{\sqrt{a}} \rightarrow \frac{1}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a}$

Examples:

$\frac{5}{\sqrt{2}} = \frac{5}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{2}}{2}$

$\frac{6}{\sqrt{3}} = \frac{6}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$

With binomial denominators, use conjugate:

$\frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}$

Converting Between Forms

Radical to Exponent

$\sqrt{x} = x^{\frac{1}{2}}$ $\sqrt[3]{x^2} = x^{\frac{2}{3}}$ $\frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}$

Exponent to Radical

$x^{\frac{3}{4}} = \sqrt[4]{x^3}$ $x^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{x}}$

Why convert? Sometimes easier to use exponent rules, sometimes radical form is clearer

SAT Question Types

Type 1: Simplify Expressions

Strategy:

• Apply exponent/radical rules step by step
• Combine like terms
• Simplify radicals completely

Type 2: Solve Equations with Exponents

Example: $2^{x+1} = 32$

Solution:

• Rewrite 32 as power of 2: $32 = 2^5$
• $2^{x+1} = 2^5$
• Same base → exponents equal: $x + 1 = 5$
• $x = 4$

Type 3: Solve Equations with Radicals

Example: $\sqrt{x + 3} = 5$

Solution:

• Square both sides: $x + 3 = 25$
• Solve: $x = 22$
• Always check! $\sqrt{22 + 3} = \sqrt{25} = 5$ ✓

Type 4: Simplify Radicals

Example: Simplify $\sqrt{200}$

Solution:

• $200 = 100 \cdot 2$
• $\sqrt{200} = \sqrt{100} \cdot \sqrt{2} = 10\sqrt{2}$

Type 5: Compare Values

Which is larger: $2^{100}$ or $4^{50}$?

Solution:

• Rewrite with same base: $4^{50} = (2^2)^{50} = 2^{100}$
• They're equal!

Common SAT Mistakes

❌ Confusing addition and multiplication of exponents
$x^2 \cdot x^3 = x^5$ (NOT $x^6$)
$(x^2)^3 = x^6$ (NOT $x^5$)

❌ Distributing exponents incorrectly
$(x + y)^2 \neq x^2 + y^2$
Must use FOIL: $(x + y)^2 = x^2 + 2xy + y^2$

❌ Forgetting to flip negative exponents
$x^{-2} = \frac{1}{x^2}$ (NOT $-x^2$)

❌ Not simplifying radicals completely
$\sqrt{48} = 4\sqrt{3}$ (NOT $2\sqrt{12}$)

❌ Squaring both sides creates extraneous solutions
Always check answers in original equation!

❌ Mishandling fractional exponents
$x^{\frac{2}{3}} = (\sqrt[3]{x})^2$ (root THEN power)

Quick Reference Table

| Expression | Equivalent | Example | |------------|-----------|---------| | $x^0$ | $1$ | $5^0 = 1$ | | $x^{-a}$ | $\frac{1}{x^a}$ | $x^{-3} = \frac{1}{x^3}$ | | $x^{\frac{1}{2}}$ | $\sqrt{x}$ | $16^{\frac{1}{2}} = 4$ | | $x^{\frac{1}{n}}$ | $\sqrt[n]{x}$ | $8^{\frac{1}{3}} = 2$ | | $x^{\frac{m}{n}}$ | $(\sqrt[n]{x})^m$ | $8^{\frac{2}{3}} = 4$ | | $x^a \cdot x^b$ | $x^{a+b}$ | $x^2 \cdot x^3 = x^5$ | | $\frac{x^a}{x^b}$ | $x^{a-b}$ | $\frac{x^5}{x^2} = x^3$ | | $(x^a)^b$ | $x^{ab}$ | $(x^2)^3 = x^6$ |

Practice Tips

✓ Memorize the rules — they appear on EVERY SAT
✓ Rewrite with same base when comparing or solving
✓ Convert between forms (radical ↔ exponent) for easier manipulation
✓ Factor out perfect squares when simplifying radicals
✓ Check solutions especially after squaring both sides
✓ Use calculator strategically for numerical values, but show algebraic work

Remember: Exponent and radical rules are fundamental — master these and many SAT problems become much easier!