6.2 Exponents, Scientific Notation & Square Roots / Radicals
Key Takeaways
- According to exponent laws, multiplying terms with identical bases requires adding exponents (a^m * a^n = a^(m+n)), while dividing terms with identical bases requires subtracting exponents (a^m / a^n = a^(m-n)).
- Any non-zero base raised to the power of zero equals 1 (a^0 = 1), and a negative exponent represents a reciprocal (a^(-n) = 1 / a^n).
- Scientific notation expresses numbers in the form N * 10^k, where 1 <= |N| < 10 and k is an integer; positive powers indicate numbers > 10, while negative powers indicate decimals < 1.
- Simplifying radicals requires factoring out perfect square components from beneath the radical sign (√(a * b) = √a * √b) and rationalizing denominators to eliminate square roots from bottom positions.
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.
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:
| Exponent Law | Algebraic Rule | Numerical / Algebraic Example |
|---|---|---|
| Product Rule | $a^m \cdot a^n = a^{m+n}$ | $x^4 \cdot x^5 = x^{4+5} = x^9$ |
| Quotient Rule | $\frac{a^m}{a^n} = a^{m-n} \quad (a \neq 0)$ | $\frac{y^9}{y^3} = y^{9-3} = y^6$ |
| Power of a Power | $(a^m)^n = a^{m \cdot n}$ | $(x^3)^4 = x^{3 \cdot 4} = x^{12}$ |
| Power of a Product | $(ab)^n = a^n b^n$ | $(3x)^3 = 3^3 \cdot x^3 = 27x^3$ |
| Power of a Quotient | $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \neq 0)$ | $\left(\frac{x}{4}\right)^2 = \frac{x^2}{4^2} = \frac{x^2}{16}$ |
| Zero Exponent Rule | $a^0 = 1 \quad (a \neq 0)$ | $(-7)^0 = 1 \quad \text{and} \quad 5x^0 = 5(1) = 5$ |
| Negative Exponent Rule | $a^{-n} = \frac{1}{a^n} \quad \text{and} \quad \frac{1}{a^{-n}} = a^n$ | $x^{-4} = \frac{1}{x^4} \quad \text{and} \quad \frac{3}{y^{-2}} = 3y^2$ |
| Rational (Fractional) Exponent | $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$ | $16^{3/4} = (\sqrt[4]{16})^3 = (2)^3 = 8$ |
Multi-Step Exponent Simplification Example
Simplify the expression completely: $\frac{(2x^4 y^{-3})^3}{4x^{-2} y^5}$
- Apply Power of a Product Rule to numerator:
- Substitute numerator back into fraction:
- Divide numerical coefficients and apply Quotient Rule to variables ($a^{m-n}$):
- Combine all simplified components: $\frac{2 x^{14}}{y^{14}}$
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 $k$.
- 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 $k$.
-
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.
- Division: Divide coefficients, subtract exponents.
- Addition & Subtraction: Rewrite terms so they share identical powers of ten before combining coefficients.
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:
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}$$Simplify the algebraic expression completely: ((3x²y⁻³)² / (9x⁻⁴y²)).
What is the result of multiplying (4.2 × 10⁵) × (3.0 × 10⁻⁸) expressed in standard scientific notation?
Simplify the radical expression: 2√50 - 3√18 + √98.