3Simplifying Expressions with Integral Exponents Helpful Tip: Watch the order of operations ….or else!Examples
4Sect 11.1 : Simplifying Expressions with Integral Exponents MAT 205 FALL 2008Sect 11.1 : Simplifying Expressions with Integral ExponentsUse one or a combination of the laws of exponents to simplify each expression. State final answers using only positive exponents. Assume all variables represent positive values.
16Fractional (Rational) Exponents When evaluating expressions involving fractional exponents without a calculator: It is usually best to find the root first, as indicated by the denominator, then raise it to the power indicated in the numerator.
17Fractional (Rational) Exponents You can evaluate expressions involving fractional exponents using the calculator. Use the key and type parentheses around the fraction.Example: Evaluate using the TI-84.
18Simplifying Expressions Involving Fractional Exponents The same laws of exponents apply to fractional exponents, though the work may be a little messier.
26Simplifying RadicalsTo reduce a radical to simplest form:Remove all perfect nth power factors from a radical of order n.If a fraction appears under the radical or there is a radical in the denominator of the expression, simplify by rationalizing the denominator.If possible, reduce the order of the radical.
27Simplifying RadicalsRemoving all nth power factors
28Simplifying RadicalsReducing the order of the radical
29Simplifying RadicalsRationalizing the denominator
32Sect 11.4: Addition and Subtraction of Radicals To perform addition or subtraction of radicals, you must have like (similar) radicals.Similar radicals differ only in their numerical coefficients (same radicand AND same index).
33Addition and Subtraction of Radicals Plan of Action:Express each radical in simplest form.Combine like radicals.Examples
37Sect 11.5: Multiplication & Division of Radicals To multiply expressions containing radicals, we will use the propertywhere a and b represent positive values.Notice that the orders (indexes) of the radicals being multiplied must be the same.
40Division of RadicalsWe saw earlier that when dealing with an expression containing a radical in the denominator, we usually rationalize the denominator.We will now deal with denominators containing two terms.
41Division of RadicalsSo, to rationalize a denominator that is the sum or difference of two terms, multiply the numerator and denominator of the fraction by the _____________________ of the denominator.
42Sect 14.4 Solving Radical Equations To solve radical equations, we will use the fact thatThat is, we will apply the appropriate inverse operation to “get the variable out of the radical”.
43Solving Radical Equations To solve a radical equation involving one radical:Isolate the radical expression on one side of the equation.Raise both sides of the equation to the power that is the same as the order of the radical (inverse operation).Solve the resulting equation for the variable.Check for extraneous solutions* by checking the apparent solutions in the original equation.*Extraneous solutions may be introduced when both sides of an equation are raised to an even power.
47To solve a radical equation involving two square roots: Isolate one of the radical expressions on one side of the equation.Square both sides of the equation.Simplify.Isolate the remaining radical expression on one side of the equation.Solve the resulting equation for the variable.Check for extraneous solutions by checking the apparent solutions in the original equation.
48Solving Radical Equations Involving Two Radicals
49Solving Radical Equations Involving Two Radicals
50Solving Radical Equations WIND POWER The power generated by a windmill is related to the velocity of the wind by the formulawhere P is the power (in watts) and v is the velocity of the wind (in mph). Find how much power the windmill is generating when the wind is 29 mph.