6.2 Exponents, Scientific Notation & Square Roots / Radicals

6.2 Exponents, Scientific Notation & Square Roots / Radicals

Exponential notation, scientific notation, and radicals are fundamental algebraic concepts tested heavily on the AFQT Mathematics Knowledge subtest. Mastering the rules governing exponents and roots allows you to simplify complex expressions quickly and compute with extremely large or small numbers without calculators.

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Laws of Exponents

An exponential expression $a^n$ consists of a base ($a$) raised to an exponent or power ($n$), representing $a$ multiplied by itself $n$ times. The eight universal laws of exponents must be memorized:

Multi-Step Exponent Simplification Example

Simplify the expression completely: $\frac{(2x^4 y^{-3})^3}{4x^{-2} y^5}$

1. Apply Power of a Product Rule to numerator: $$(2x^4 y^{-3})^3 = 2^3 \cdot (x^4)^3 \cdot (y^{-3})^3 = 8 x^{12} y^{-9}$$
2. Substitute numerator back into fraction: $$\frac{8 x^{12} y^{-9}}{4 x^{-2} y^5}$$
3. Divide numerical coefficients and apply Quotient Rule to variables ($a^{m-n}$): $$\text{Coefficients: } \frac{8}{4} = 2$$ $$\text{Variable } x: x^{12 - (-2)} = x^{12 + 2} = x^{14}$$ $$\text{Variable } y: y^{-9 - 5} = y^{-14} = \frac{1}{y^{14}}$$
4. Combine all simplified components: $\frac{2 x^{14}}{y^{14}}$

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Scientific Notation

Scientific notation expresses numbers as a product of a decimal coefficient $N$ and a power of ten: $N \times 10^k$, where $1 \le |N| < 10$ and $k$ is an integer.

Converting Between Formats

• Standard to Scientific:

  • For numbers $\ge 10$: Move the decimal point to the left until one non-zero digit remains to the left of the decimal. The number of places moved is a positive exponent $k$. $$456,000 = 4.56 \times 10^5$$
  • For decimals $< 1$: Move the decimal point to the right until one non-zero digit is to the left. The number of places moved is a negative exponent $k$. $$0.000078 = 7.8 \times 10^{-5}$$

• Scientific to Standard:

  • If $k > 0$, shift decimal $k$ places to the right (adding zeros as needed).
  • If $k < 0$, shift decimal $|k|$ places to the left.

Arithmetic Operations with Scientific Notation

• Multiplication: Multiply coefficients, add exponents. $$(3.0 \times 10^4) \times (4.5 \times 10^6) = (3.0 \times 4.5) \times 10^{4+6} = 13.5 \times 10^{10} = 1.35 \times 10^{11}$$
• Division: Divide coefficients, subtract exponents. $$(8.4 \times 10^7) \div (2.1 \times 10^{-3}) = (8.4 \div 2.1) \times 10^{7 - (-3)} = 4.0 \times 10^{10}$$
• Addition & Subtraction: Rewrite terms so they share identical powers of ten before combining coefficients. $$(5.2 \times 10^5) + (3.1 \times 10^4) = (5.2 \times 10^5) + (0.31 \times 10^5) = (5.2 + 0.31) \times 10^5 = 5.51 \times 10^5$$

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Square Roots and Radicals

The principal square root $\sqrt{x}$ represents the non-negative real number that, when multiplied by itself, equals $x$.

Perfect Squares Reference Table

Memorizing perfect square integers accelerates AFQT radical operations:

$$\begin{array}{ccccc} 1^2 = 1 & 2^2 = 4 & 3^2 = 9 & 4^2 = 16 & 5^2 = 25 \\ 6^2 = 36 & 7^2 = 49 & 8^2 = 64 & 9^2 = 81 & 10^2 = 100 \\ 11^2 = 121 & 12^2 = 144 & 13^2 = 169 & 14^2 = 196 & 15^2 = 225 \end{array}$$

Radical Properties & Simplification Rules

1. Product Property of Radicals: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ To simplify $\sqrt{N}$, factor $N$ into the largest perfect square factor ($S$) and a non-square factor ($R$): $\sqrt{N} = \sqrt{S \cdot R} = \sqrt{S} \cdot \sqrt{R}$.

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

2. Quotient Property of Radicals: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \quad (b > 0)$

   $$\text{Example: } \sqrt{\frac{25}{9}} = \frac{\sqrt{25}}{\sqrt{9}} = \frac{5}{3}$$

3. Combining Like Radicals: Radicals can only be added or subtracted if they share the exact same radicand.

   $$\text{Simplify: } 3\sqrt{12} + 5\sqrt{27}$$
   • Step 1: Simplify each term: $3\sqrt{12} = 3\sqrt{4 \times 3} = 3(2\sqrt{3}) = 6\sqrt{3}$
   • Step 2: $5\sqrt{27} = 5\sqrt{9 \times 3} = 5(3\sqrt{3}) = 15\sqrt{3}$
   • Step 3: Combine like terms: $6\sqrt{3} + 15\sqrt{3} = 21\sqrt{3}$

4. Rationalizing the Denominator: Standard mathematical form prohibits leaving a radical in the denominator of a fraction.

   • Single Radical Denominator: Multiply numerator and denominator by the radical. $$\frac{6}{\sqrt{3}} = \frac{6 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$$
   • Binomial Radical Denominator: Multiply numerator and denominator by the conjugate ($(a + \sqrt{b}) \leftrightarrow (a - \sqrt{b})$). $$\frac{4}{3 - \sqrt{5}} = \frac{4(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})} = \frac{12 + 4\sqrt{5}}{3^2 - (\sqrt{5})^2} = \frac{12 + 4\sqrt{5}}{9 - 5} = \frac{12 + 4\sqrt{5}}{4} = 3 + \sqrt{5}$$