SAT Passport to Advanced Math: Exponents & Radicals

This is the second part of our guide to SAT Number Properties. Here, we focus on the algebraic tools you'll need for the "Passport to Advanced Math" domain, including the laws of exponents and the crucial skill of working with radicals (square roots).

Section 1: Fundamentals of Exponents (Indices)

The Foundation: What is an Exponent?

Start with the core definitions, including positive, negative, zero, and fractional exponents (roots). Build an intuitive understanding with our interactive explorer.

Repeated Multiplication

$$a^n$$ means multiplying 'a' by itself 'n' times.

Core Application: Properties & Laws of Exponents

Master the essential rules for manipulating exponents, such as the product, quotient, and power rules. These are critical for simplifying complex expressions on the SAT.

Basic Operations

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

Verify with numbers:

$$ a^m / a^n $$ = $$ a^{m-n} $$

Verify with numbers:

Section 2: SAT-Specific Skills & Formulas

Key SAT Skill: Working with Radicals

Learn the essential techniques for simplifying radicals (e.g., √72) and combining like terms, a common task in the no-calculator section.

Step 1Simplify $$ \sqrt{72} $$

The goal is to find the largest perfect square factor.

Step 2Find the Largest Perfect Square Factor

The perfect squares are 4, 9, 16, 25, 36...
The largest one that divides 72 is 36.

Step 3Rewrite the Radical

$$ \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} $$

Step 4Simplify the Perfect Square

$$ \sqrt{36} \times \sqrt{2} = 6 \times \sqrt{2} = 6\sqrt{2} $$

Essential Tool: Key Algebraic Identities

Memorize and apply the most important algebraic formulae, like (a+b)² and a²-b², which are frequently used to manipulate and simplify expressions on the SAT.

Basic Formulae for Squares and Cubes

$$ (a+b)^2 $$ = $$ a^2 + b^2 + 2ab $$

Verify with numbers:

$$ (a-b)^2 $$ = $$ a^2 + b^2 - 2ab $$

Verify with numbers:

$$ a^2 - b^2 $$ = $$ (a+b)(a-b) $$

Verify with numbers:

$$ (a+b)^3 $$ = $$ a^3 + b^3 + 3ab(a+b) $$

Verify with numbers:

$$ (a-b)^3 $$ = $$ a^3 - b^3 - 3ab(a-b) $$

Verify with numbers:

$$ a^3 + b^3 $$ = $$ (a+b)(a^2 - ab + b^2) $$

Verify with numbers:

$$ a^3 - b^3 $$ = $$ (a-b)(a^2 + ab + b^2) $$

Verify with numbers:

Interactive Practice

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