2.3 Exponents, Roots & Absolute Value

2.3 Exponents, Roots & Absolute Value

Exponents, roots, and absolute values appear pervasively throughout algebra, geometry, and numerical estimation on the GRE. Understanding their algebraic properties and geometric interpretations will allow you to simplify complicated expressions quickly.

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Rules of Exponents

An exponent indicates how many times a base number $x$ is multiplied by itself ($x^n = x \times x \times \dots \times x$).

Primary Exponent Laws

Common GRE Trap: Do NOT add bases when adding exponential terms! $2^5 + 2^5 = 2 \times (2^5) = 2^1 \times 2^5 = 2^6$. It is NOT $4^5$ or $2^{10}$!

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Radicals & Fractional Exponents

A fractional exponent represents a radical root: $x^{\frac{m}{n}} = \sqrt[n]{x^m} = \left(\sqrt[n]{x}\right)^m$ where $n$ is the root index and $m$ is the power.

Simplifying Radicals

To simplify a square root $\sqrt{N}$, factor $N$ into a product of its largest perfect square factor and a remaining factor: $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$

Examples of Radical Simplification:

• $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$
• $\sqrt{18} + \sqrt{50} = \sqrt{9 \times 2} + \sqrt{25 \times 2} = 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$

Rationalizing Denominators

On the GRE, options are typically written without radicals in the denominator. To rationalize $\frac{a}{\sqrt{b}}$, multiply numerator and denominator by $\sqrt{b}$: $\frac{1}{\sqrt{3}} = \frac{1 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{3}$ For binomial denominators, multiply by the conjugate: $\frac{1}{3 + \sqrt{2}} = \frac{1(3 - \sqrt{2})}{(3 + \sqrt{2})(3 - \sqrt{2})} = \frac{3 - \sqrt{2}}{3^2 - (\sqrt{2})^2} = \frac{3 - \sqrt{2}}{9 - 2} = \frac{3 - \sqrt{2}}{7}$

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Absolute Value Definitions & Properties

The absolute value of a real number $x$, denoted $|x|$, measures its magnitude or distance from zero on the real number line regardless of direction.

Formal Algebraic Definition

$|x| = \begin{cases} x & \text{if } x \ge 0 \ -x & \text{if } x < 0 \end{cases}$

Key Concept: $\sqrt{x^2} = |x|$. For instance, $\sqrt{(-5)^2} = \sqrt{25} = 5 = |-5|$.

Core Properties of Absolute Value

1. Non-negativity: $|x| \ge 0$ for all real numbers $x$.
2. Product Property: $|a \cdot b| = |a| \cdot |b|$
3. Quotient Property: $\left|\frac{a}{b}\right| = \frac{|a|}{|b|}$ ($b eq 0$)
4. Triangle Inequality: $|a + b| \le |a| + |b|$

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Absolute Value Equations & Inequalities

Solving Absolute Value Equations

To solve $|E(x)| = c$ where $c > 0$, split into two separate equations: $E(x) = c \quad \text{OR} \quad E(x) = -c$

Worked Example: Solve $|2x - 5| = 9$

• Case 1: $2x - 5 = 9 \implies 2x = 14 \implies x = 7$
• Case 2: $2x - 5 = -9 \implies 2x = -4 \implies x = -2$
• Solutions: $x = 7$ or $x = -2$.

Solving Absolute Value Inequalities

Worked Example: Solve $|3x - 4| \le 11$

1. Unfold into compound inequality: $-11 \le 3x - 4 \le 11$
2. Add $4$ to all three parts: $-7 \le 3x \le 15$
3. Divide all three parts by $3$: $-\frac{7}{3} \le x \le 5$