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Revision of Exponent Laws

• Exponent laws are used to simplify exponential expressions, ensuring they are written with the same base whenever possible.
• The conditions for these laws are ( a > 0 ), ( b > 0 ), and ( m, n \in \mathbb{R} ).
• Multiplication of exponents with the same base: ( a^m \times a^n = a^{m+n} )
• Division of exponents with the same base: ( \frac{a^m}{a^n} = a^{m-n} )
• Raising a product to a power: ( (ab)^n = a^n b^n )
• Raising a quotient to a power: ( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} )
• Power of a power: ( (a^m)^n = a^{mn} )
• Zero exponent: ( a^0 = 1 ) (for any ( a \neq 0 ))
• Negative exponent: ( a^{-n} = \frac{1}{a^n} )

Factorisation of Exponential Expressions

• Common Factors: Factor out the greatest common factor (GCF) in the expression.
• Difference of Squares: Apply the formula ( a^2 - b^2 = (a - b)(a + b) ).
• Prime Factorisation: Rewrite the bases in terms of their prime factors to simplify the expressions.

Simplification of Exponential Expressions

• Rewrite all terms with the same base, converting to prime bases if necessary.
• Apply the exponent laws to combine or reduce the exponents using multiplication, division, and power rules.
• Factorise the expression to cancel common factors in the numerator and denominator.
• Simplify complex fractions by changing to prime bases, factorising, and then simplifying by cancelling common terms.

Rational Exponents

• Rational exponents apply the same laws of exponents to expressions where the exponents are fractions.
• Multiplication of exponents: ( a^{m/n} \times a^{p/q} = a^{m/n + p/q} )
• Division of exponents: ( \frac{a^{m/n}}{a^{p/q}} = a^{m/n - p/q} )
• Power of a Power: ( \left(a^{m/n}\right)^{p/q} = a^{(m/n) \cdot (p/q)} = a^{\frac{mp}{nq}} )
• Raising a Product to a Power: ( (ab)^{m/n} = a^{m/n} b^{m/n} )
• Raising a Quotient to a Power: ( \left(\frac{a}{b}\right)^{m/n} = \frac{a^{m/n}}{b^{m/n}} )

Simplifying Rational Exponents

• Convert roots to fractional exponents (e.g., ( \sqrt[n]{a} = a^{1/n} )).
• Use the laws of exponents to combine and simplify the expressions.
• Simplify the fractions in the exponents as needed.

Exponential Equations

• Exponential equations have the variable in the exponent.
• If ( a^x = a^y ), then ( x = y ) (assuming ( a > 0 ) and ( a \neq 1 )).
• Equating Exponents: Write both sides of the equation with the same base, then set the exponents equal to each other.
• Using Logarithms: Use logarithms to solve the equation when bases cannot easily be made the same, isolating the exponent.

Steps to Solve Exponential Equations

• Rewrite the equation so both sides have the same base, if possible.
• Set the exponents equal to each other once the bases are the same, and solve for the variable.
• Verify the solutions in the original equation to ensure they are valid and not extraneous.

Solving Linear Equations

• A linear equation has a highest variable exponent of 1, with at most one solution.
• To solve: expand expressions, group terms, and factorise.
• Expand all brackets to simplify both sides of the equation.
• Rearrange the terms to move all variable terms to one side and constants to the other.
• Group like terms together to simplify the equation.
• Factorise if necessary, factoring out common terms.
• Solve for the variable by isolating it.
• Substitute the solution back into the original equation to check the answer.
• Maintain balance by performing the same operation on both sides of the equation.

Solving Quadratic Equations

• A quadratic equation has a maximum variable exponent of 2, with at most two solutions, but can have one or none.
• Rewrite the equation in the form ( ax^2 + bx + c = 0 ).
• Divide by common factors to simplify, if applicable.
• Factorise the quadratic expression ( ax^2 + bx + c = 0 ) into the form ( (rx + s)(ux + v) = 0 ).
• Solve for both factors: ( rx + s = 0 ) and ( ux + v = 0 ).
• Check the solution by substituting back into the original equation.
• Keep the equation balanced by performing any operation to both sides.

Solving Simultaneous Equations

• Simultaneous equations involve solving for two unknown variables using two equations.
• The solutions are the values that satisfy both equations simultaneously.

Solving by Substitution

• Express one variable in terms of the other using the simplest equation.
• Substitute into the second equation to reduce the number of variables by one.
• Solve the resulting equation with one unknown variable.
• Substitute the solution back into the first equation to find the value of the other variable.

Solving by Elimination

• Make the coefficients of one variable the same in both equations.
• Add or subtract the equations to eliminate that variable.
• Solve the resulting equation for the remaining variable.
• Substitute the value back into one of the original equations to find the eliminated variable's value.

Solving Graphically

• The solution to simultaneous equations graphically are the coordinates where the two graphs intersect.

Word Problems

• Translate word problems into mathematical equations to find a solution.
• Read the whole question to understand the problem.
• Determine what you are asked to solve for.
• Assign a variable, like ( x ), to the unknown quantity.
• Translate words into algebraic expressions using the variable.
• Set up an equation or system of equations.
• Solve the equation algebraically using substitution.
• Check the solution to ensure it makes sense in the context of the problem.

Literal Equations

• Literal equations have multiple letters or variables, like ( A = \pi r^2 ) or ( v = \frac{D}{t} ).
• Solve for one variable by isolating it, asking what it is joined to and how, then performing the opposite operation on both sides.
• If the unknown variable is in two or more terms, factor it out as a common factor.
• When taking the square root of both sides, remember both positive and negative answers.
• If the unknown variable is in the denominator, multiply both sides by the lowest common denominator (LCD) and continue to solve.

Solving Linear Inequalities

• Linear inequalities are similar to linear equations, where the largest exponent of a variable is 1. Examples: ( 2x + 2 \leq 1 ), ( 2 - x \geq 3 ), etc.
• Use similar methods as solving linear equations.
• When multiplying or dividing by a negative number, switch the inequality sign.