Exponents and Radicals MAT 205 FALL 2008 Exponents and Radicals

This symbol is the radical or the radical sign Radical Expressions Finding a root of a number is the inverse operation of raising a number to a power. This symbol is the radical or the radical sign radical sign index radicand The expression under the radical sign is the radicand. The index defines the root to be taken.

Fractional (Rational) Exponents Note: These properties are valid as long as does not involve the even root of a negative number. power root

Fractions as exponents Under Radical (exponent) Outside Radical (root)

Fractions as exponents as exponent means as exponent means as exponent means as exponent means

An nth root of any number a is a number whose nth power is a. nth Roots An nth root of any number a is a number whose nth power is a. Examples:

Examples:

Negative Exponents: Remember that a negative in the exponent does not make the number negative! If a base has a negative exponent, that indicates it is in the “wrong” position in fraction. That base can be moved across the fraction bar and given a positive exponent. EXAMPLES:

or or Examples: or

In general, a radical expression is simplified when: The radicand contains no fractions. No radicals appear in the denominator = (Rationalization) The radicand contains no factors that are nth powers of an integer or polynomial.

Use the properties of exponents to simplify each expression

Example

Simplifying Radicals Let a and b represent positive real numbers.

Simplifying Radicals Remember:

Simplify Radical expression

Simplify.

Simplify

simplify

Product Property of Radicals Factor into cubes if possible Product Property of Radicals

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.

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

Rationalizing

This cannot be divided which leaves the radical in the denominator This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. 42 cannot be simplified, so we are finished.

Simplify

Rationalize the denominator Simplify each expression. Rationalize the denominator Answer

Rationalizing the Denominator of Radicals Expressions

Rationalizing the Denominator of Radicals Expressions If the denominator contains a radical and it is not a monomial term, then the use of a conjugate is required. conjugate

Simplify Use the conjugate

Rationalizing the Denominator of Radicals Expressions conjugate

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.

Radical Equations and Problem Solving

EXAMPLE 2 2 Square both sides to get rid of the square root ( )

RADICAL EQUATIONS ISOLATE RADICAL / RATIONAL RAISE BOTH SIDES TO RECIPROCAL POWER SOLVE FOR THE VARIABLE

Radical Equations and Problem Solving

Radical Equations and Problem Solving

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.

EXAMPLE ( ) 2 2 NO SOLUTION Since 16 doesn’t plug in as a solution. Note: You will get Extraneous Solutions from time to time – always do a quick check Let’s Double Check that this works

Radical Equations and Problem Solving

Radical Equations and Problem Solving

Radical Equations and Problem Solving

Check: n=2 Since the root of x=2 does not check, it is called an extraneous solution.

Check #2: n = 27 The solution is n=27.

Check: y=3

Solving Radical Equations Set up the equation so that there will be only one radical on each side of the equal sign. Solve Square both sides of the equation. Use Foil. Simplify. Simplify by dividing by a common factor of 2. Square both sides of the equation. Use Foil.

x - 3 = 0 or x - 7 = 0 x = 3 or x = 7 Solving Radical Equations Distribute the 4. Simplify. Factor the quadratic. Solve for x. x - 3 = 0 or x - 7 = 0 x = 3 or x = 7 Verify both solutions. L.S. R.S. L.S. R.S.