Quotient Rule

$\frac{a^m}{a^n} = a^{m-n}$

Power Rule

$(a^m)^n = a^{mn}$

Negative Exponents

$a^{-n} = \frac{1}{a^n}$

Zero Exponent

$a^0 = 1$ (if $a \neq 0$)

Radicals

Simplifying

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

Converting Between Forms

$\sqrt[n]{a^m} = a^{m/n}$

Examples:

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

SAT Tricks

• Fractional exponents: $x^{3/2} = (\sqrt{x})^3 = \sqrt{x^3}$
• Rationalizing: $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
• Watch for: $(2x)^3 = 8x^3$, not $2x^3$!

Answer: $x^5$

SAT Tip: Apply power rule before quotient rule!

$\frac{x^5}{x^2} = x^{5-2} = x^3$

Answer: $x^3$

$\frac{x^5}{x^2} = x^{5-2} = x^3$

Answer: $x^3$

$\sqrt[3]{x^5} = x^{5/3}$

Check: $x^{5/3} = x^{1 + 2/3} = x \cdot x^{2/3} = x\sqrt[3]{x^2}$

Answer: $x^{5/3}$

SAT Tip: The SAT frequently asks you to convert between radical and exponential notation.

$\sqrt[3]{x^5} = x^{5/3}$

Check: $x^{5/3} = x^{1 + 2/3} = x \cdot x^{2/3} = x\sqrt[3]{x^2}$

Answer: $x^{5/3}$

SAT Tip: The SAT frequently asks you to convert between radical and exponential notation.

Set the exponents equal (same base): $3x = 2x + 2$ $x = 2$

Check: $27^2 = 729$ and $9^3 = 729$ ✓

Answer: $x = 2$

Strategy: When you have exponential equations, try to make the bases the same, then set exponents equal.

Set the exponents equal (same base): $3x = 2x + 2$ $x = 2$

Check: $27^2 = 729$ and $9^3 = 729$ ✓

Answer: $x = 2$

Strategy: When you have exponential equations, try to make the bases the same, then set exponents equal.

Step 2: Write the full fraction: $\frac{27x^6y^{-3}}{9x^{-1}y^4}$

Step 3: Simplify coefficients: $\frac{27}{9} = 3$

Step 4: Apply quotient rule for each variable: $x^{6-(-1)} = x^7$ $y^{-3-4} = y^{-7} = \frac{1}{y^7}$

Step 5: Combine: $3 \cdot x^7 \cdot y^{-7} = \frac{3x^7}{y^7}$

Answer: $\frac{3x^7}{y^7}$

Step 2: Write the full fraction: $\frac{27x^6y^{-3}}{9x^{-1}y^4}$

Step 3: Simplify coefficients: $\frac{27}{9} = 3$

Step 4: Apply quotient rule for each variable: $x^{6-(-1)} = x^7$ $y^{-3-4} = y^{-7} = \frac{1}{y^7}$

Step 5: Combine: $3 \cdot x^7 \cdot y^{-7} = \frac{3x^7}{y^7}$

Answer: $\frac{3x^7}{y^7}$