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Rational Exponents and Surds

• Laws of exponents extend to rational numbers, which can be expressed as a fraction with integers in the numerator and denominator.
• If $r^n = a$, then $r = \sqrt[n]{a}$ for $n \geq 2$.
• $a^{\frac{1}{n}} = \sqrt[n]{a}$
• $a^{-\frac{1}{n}} = (a^{-1})^{\frac{1}{n}} = \frac{1}{\sqrt[n]{a}}$
• $a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m}$ where $a > 0$, $r > 0$, and $m, n \in \mathbb{Z}$, $n \neq 0$.

Definitions and Properties

• The square root of 25 is $\sqrt{25} = 5$.
• The cube root of 8 is $\sqrt[3]{8} = 2$.
• The fifth root of 32 is $\sqrt[5]{32} = 2$.

Radical Symbols

• The radical symbol $\sqrt[n]{a}$ indicates the $n$-th root of $a$.
• If $n$ is even, the radicand must be positive.
• If $n$ is odd, the radicand can be positive or negative.

Surds

• A surd is a radical resulting in an irrational number.
• Examples of surds include $\sqrt{12}$, $\sqrt{100}$, and $\sqrt{25}$.

Simplification of Surds

• $\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab}$
• $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$
• $\sqrt[n]{a^m} = a^{\frac{m}{n}}$
• $(\sqrt[n]{a})^m = a^{\frac{m}{n}}$

Rationalising Denominators

• Rationalising the denominator means eliminating the surd from the denominator.
• Achieved by multiplying the numerator and denominator by the conjugate of the denominator or another suitable expression.

Laws of Exponents

• Product of Powers: $a^m \times a^n = a^{m+n}$
• Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$
• Power of a Power: $(a^m)^n = a^{mn}$
• Negative Exponents: $a^{-m} = \frac{1}{a^m}$
• Fractional Exponents: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$
• Zero Exponents: $a^0 = 1$ (for $a \neq 0$)

Summary

• Rational Exponents: Represent roots and powers using fractional exponents.
• Surds: Radicals that result in irrational numbers
• Simplifying Surds: Use properties of exponents and roots to simplify expressions.
• Rationalising Denominators: Eliminate radicals from the denominator for easier manipulation.
• Exponent Laws: Rules that govern the manipulation of expressions involving exponents.

Solving Surd Equations

• Isolate the surd on one side of the equation and eliminate it by raising both sides to the power corresponding to the root.

Steps for Solving Surd Equations

• Write in Exponential Notation: Convert surds into exponential form to ease manipulation.
• Divide Both Sides and Simplify: Simplify the equation by isolating the variable term.
• Simplify the Exponents: Apply exponent rules to simplify the terms.
• Factorize if Necessary: Factorize the equation if it is quadratic in form.
• Solve for Both Factors: Use the zero-product property to solve for the variable.
• Check the Solution: Substitute back into the original equation to verify.

Key Points to Remember

• Isolate the surd, ensuring it is the only term on one side of the equation.
• When raising both sides to a power, be cautious of extraneous solutions.
• Always check solutions by substituting back into the original equation.
• Factorize and solve for the variable if the equation is quadratic.

Solving Quadratic Surd Equations

• Isolate and Simplify: Get the surd term by itself and simplify the equation.
• Square Both Sides: Raise both sides to the necessary power to eliminate the surd.
• Solve the Quadratic Equation: Solve for the variable using algebraic methods.
• Check Solutions: Verify the solutions by substituting them back.

Applications of Exponentials

• Exponentials are used to model real-world phenomena like population growth, radioactive decay, and compound interest.

Exponential Growth and Decay

• Exponential growth model: quantity increases by a fixed percentage each period.
• Exponential decay model: quantity decreases by a fixed percentage each period.
• General formula for exponential growth or decay: $P(t) = P_0 \times (1 + r)^t$
  • $P(t)$ is the final amount after time t.
  • $P_0$ is the initial amount.
  • $r$ is the growth rate (positive for growth, negative for decay).
  • $t$ is the time period.

Population Growth

• The formula can be used to predict the future population based on current population and the growth rate.

Compound Interest

• Compound interest formula is similar to exponential growth.
• Used to calculate accumulated money over time with a given interest rate.

Solving Exponential Problems

• Identify the variables: Determine the initial amount ($P_0$), growth rate ($r$), and time period ($t$).
• Apply the formula: Use the exponential growth or decay formula to find the final amount.
• Convert percentages: Convert percentage rates into decimal form for use in the formula.

Summary of Exponential Rules

• The number system:
  • $\mathbb{N}$: Natural numbers {1, 2, 3,...}
  • $\mathbb{N}_0$: Whole numbers {0, 1, 2, 3,...}
  • $\mathbb{Z}$: Integers {..., -3, -2, -1, 0, 1, 2, 3,...}
  • $\mathbb{Q}$: Rational numbers (fractions and terminating or recurring decimals)
  • $\mathbb{Q}^c$: Irrational numbers (non-terminating, non-recurring decimals)
  • $\mathbb{R}$: Real numbers (all rational and irrational numbers)
• Definitions and Laws of Exponents:
  • $a^n = a \times a \times a \times \cdots \times a$ (n times)
  • $a^0 = 1$ (for $a \neq 0$)
  • $a^{-n} = \frac{1}{a^n}$ (for $a \neq 0$)
  • $a^m \times a^n = a^{m+n}$
  • $\frac{a^m}{a^n} = a^{m-n}$
  • $(ab)^n = a^n \times b^n$
  • $(\frac{a}{b})^n = \frac{a^n}{b^n}$
  • $(a^m)^n = a^{mn}$
• Rational Exponents and Surds:
  • $\sqrt[n]{a} = a^{\frac{1}{n}}$
  • $a^{-\frac{1}{n}} = \frac{1}{a^{\frac{1}{n}}}$
  • $a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m}$