Radicals and Integer Exponents

Rules of Exponents

| Rule | Formula | Example | |------|---------|---------| | Product | $a^m \cdot a^n = a^{m+n}$ | $x^3 \cdot x^4 = x^7$ | | Quotient | $\frac{a^m}{a^n} = a^{m-n}$ | $\frac{x^5}{x^2} = x^3$ | | Power of a power | $(a^m)^n = a^{mn}$ | $(x^3)^2 = x^6$ | | Zero exponent | $a^0 = 1$ ($a \neq 0$) | $5^0 = 1$ | | Negative exponent | $a^{-n} = \frac{1}{a^n}$ | $2^{-3} = \frac{1}{8}$ |

Scientific Notation

A number in scientific notation: $a \times 10^n$ where $1 \leq a < 10$.

Examples:

• $45,000 = 4.5 \times 10^4$
• $0.003 = 3 \times 10^{-3}$
• $6,020,000 = 6.02 \times 10^6$

Operations with Scientific Notation

Multiply: $(3 \times 10^4)(2 \times 10^5) = 6 \times 10^9$

Divide: $\frac{8 \times 10^7}{4 \times 10^3} = 2 \times 10^4$

Square Roots and Cube Roots

$\sqrt{x} = x^{1/2} \qquad \sqrt[3]{x} = x^{1/3}$

$\sqrt{36} = 6 \qquad \sqrt[3]{27} = 3 \qquad \sqrt[3]{-8} = -2$

Simplifying Square Roots

$\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$

$\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$

Tip: Find the largest perfect square factor when simplifying radicals.