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**Exponents and Radicals**

MAT 205 FALL 2008 Chapter 11 Exponents and Radicals

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**Flashback: One semester ago……..**

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**Simplifying Expressions with Integral Exponents**

Helpful Tip: Watch the order of operations ….or else! Examples

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**Sect 11.1 : Simplifying Expressions with Integral Exponents**

MAT 205 FALL 2008 Sect 11.1 : Simplifying Expressions with Integral Exponents Use 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.

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**Simplifying Expressions with Integral Exponents**

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**Simplifying Expressions with Integral Exponents**

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**Simplifying Expressions with Integral Exponents**

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**Simplifying Expressions with Integral Exponents**

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**Simplifying Expressions with Integral Exponents**

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**Simplifying Expressions with Integral Exponents**

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**Simplifying Expressions with Integral Exponents**

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**Simplifying Expressions with Integral Exponents**

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**Simplifying Expressions with Integral Exponents**

…and the calc student’s nightmare:

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P. 317 #66 The metric units for the velocity v of an object are , and the units for acceleration a of the object are What are the units for v/a?

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**Sect 11.2: Fractional (Rational) Exponents**

power root

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**Fractional (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.

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**Fractional (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.

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**Simplifying Expressions Involving Fractional Exponents**

The same laws of exponents apply to fractional exponents, though the work may be a little messier.

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**Simplifying Expressions Involving Fractional Exponents**

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**Simplifying Expressions Involving Fractional Exponents**

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**Simplifying Expressions Involving Fractional Exponents**

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**Simplifying Expressions Involving Fractional Exponents**

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**Simplifying Expressions Involving Fractional Exponents**

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**Sect 11.3: Simplifying Radicals**

Let a and b represent positive real numbers.

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Simplifying Radicals Remember ….

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Simplifying Radicals To 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.

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Simplifying Radicals Removing all nth power factors

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Simplifying Radicals Reducing the order of the radical

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Simplifying Radicals Rationalizing the denominator

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Simplifying Radicals Mixed bag…

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Simplifying Radicals Two more…

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**Sect 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).

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**Addition and Subtraction of Radicals**

Plan of Action: Express each radical in simplest form. Combine like radicals. Examples

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**Addition and Subtraction of Radicals**

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**Addition and Subtraction of Radicals**

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**Addition and Subtraction of Radicals**

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**Sect 11.5: Multiplication & Division of Radicals**

To multiply expressions containing radicals, we will use the property where a and b represent positive values. Notice that the orders (indexes) of the radicals being multiplied must be the same.

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**Multiplication of Radicals**

Check it out!

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**Multiplication of Radicals**

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Division of Radicals We 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.

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Division of Radicals So, 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.

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**Sect 14.4 Solving Radical Equations**

To solve radical equations, we will use the fact that That is, we will apply the appropriate inverse operation to “get the variable out of the radical”.

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**Solving 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.

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**Solving Radical Equations**

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**Solving Radical Equations**

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**Solving Radical Equations**

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**To 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.

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**Solving Radical Equations Involving Two Radicals**

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**Solving Radical Equations Involving Two Radicals**

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**Solving Radical Equations**

WIND POWER The power generated by a windmill is related to the velocity of the wind by the formula where 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.

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MAT 205 FALL 2008 End of Section

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