What is the simplified value of the numerical expression: ((2^3 * 3^2)^3) / (2^7 * 3^4)?

First, apply the power of a product rule to the numerator: (2^3 * 3^2)^3 = 2^(33) * 3^(23) = 2^9 * 3^6. Next, apply the quotient rule by subtracting exponents of like bases: (2^9 * 3^6) / (2^7 * 3^4) = 2^(9-7) * 3^(6-4) = 2^2 * 3^2 = 4 * 9 = 36.

Which of the following represents the simplified form of the radical expression: √48 + √75 - √108?

3√3. Simplify each radical term by factoring out perfect squares: √48 = √(16 * 3) = 4√3; √75 = √(25 * 3) = 5√3; √108 = √(36 * 3) = 6√3. Combine like radical terms: 4√3 + 5√3 - 6√3 = (4 + 5 - 6)√3 = 3√3.

What is the complete solution set for the absolute value inequality: |3x - 4| ≤ 11?

-7/3 ≤ x ≤ 5. Unfold the absolute value inequality |3x - 4| ≤ 11 into the compound inequality -11 ≤ 3x - 4 ≤ 11. Add 4 to all parts of the inequality: -7 ≤ 3x ≤ 15. Finally, divide all parts by 3 to isolate x: -7/3 ≤ x ≤ 5.

What is the exact numerical value of the expression (64)^(-2/3)?

1/16. Express (64)^(-2/3) using radical and negative exponent rules: (64)^(-2/3) = 1 / ((64)^(2/3)) = 1 / ((∛64)^2). Evaluate the cube root of 64 first: ∛64 = 4. Square the result: 4^2 = 16. Taking the reciprocal gives 1/16.

Exponents, Roots & Absolute Value — Free Study Guide 2026

Key Takeaways

Exponent product, quotient, and power rules: x^a * x^b = x^(a+b), x^a / x^b = x^(a-b), and (x^a)^b = x^(a*b).
Zero exponent x^0 = 1 (for x ≠ 0), negative exponent x^(-n) = 1 / x^n, and fractional exponent x^(m/n) = n-th root of (x^m).
To simplify radical expressions, factor out perfect nth powers: √(a * b) = √a * √b for positive real numbers (e.g., √72 = √(36 * 2) = 6√2).
The absolute value definition is piecewise: |x| = x when x ≥ 0, and |x| = -x when x < 0; geometrically, |a - b| is the distance between a and b.
For any constant b > 0, the absolute value inequality |x - a| ≤ b is equivalent to the compound inequality a - b ≤ x ≤ a + b.

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.

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

Rule	Formula	Example
Product Rule	$x^a \cdot x^b = x^{a+b}$	$2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$
Quotient Rule	$\frac{x^a}{x^b} = x^{a-b}$ ($x
eq 0$)	$\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$
Power Rule	$(x^a)^b = x^{a \cdot b}$	$(3^2)^3 = 3^{2 \cdot 3} = 3^6 = 729$
Power of a Product	$(x \cdot y)^n = x^n \cdot y^n$	$(2 \cdot 5)^3 = 2^3 \cdot 5^3 = 8 \cdot 125 = 1000$
Power of a Quotient	$\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$ ($y
eq 0$)	$\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}$
Zero Exponent	$x^0 = 1$ ($x
eq 0$)	$99^0 = 1$ ; note that $0^0$ is undefined
Negative Exponent	$x^{-n} = \frac{1}{x^n}$ ($x
eq 0$)	$4^{-2} = \frac{1}{4^2} = \frac{1}{16}$

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 $2^{10}$ !

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}

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

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

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

Inequality Form ( $b > 0$ )	Equivalent Compound Inequality	Geometric Meaning
$\vert x - a\vert\le b$	$a - b \le x \le a + b$	Distance between $x$ and $a$ is $\le b$ (Bounded interval)
$\vert x - a\vert\ge b$	$x \le a - b \quad \text{OR} \quad x \ge a + b$	Distance between $x$ and $a$ is $\ge b$ (Two outer rays)

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

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

GRE Quantitative Reasoning

2.3 Exponents, Roots & Absolute Value

Key Takeaways

2.3 Exponents, Roots & Absolute Value

Rules of Exponents

Primary Exponent Laws

Radicals & Fractional Exponents

Simplifying Radicals

Examples of Radical Simplification:

Rationalizing Denominators

Absolute Value Definitions & Properties

Formal Algebraic Definition

Core Properties of Absolute Value

Absolute Value Equations & Inequalities

Solving Absolute Value Equations

Worked Example: Solve ∣2x−5∣=9|2x - 5| = 9∣2x−5∣=9

Solving Absolute Value Inequalities

Worked Example: Solve ∣3x−4∣≤11|3x - 4| \le 11∣3x−4∣≤11

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

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