Algebra

Example 3

3/2- square root 3= 3(2+ sqr 3) / (2-sqr 3)(2+sqr 3)= 6+3sqr3/ 4-sqr9 = 6+3sqr3

The 3 conditions

A. No perfect nth power factors of a radicand can be in an nth degree radical. B. No fractions can be in the radicand. C. No radicals can be in the denominator of a fraction.

add exponents when?

Multiplying like bases

fact

Notice that if a = b, the exponent is 1; so, any base raised to the same power as the index will equal the base.

lesson number one

Transcripts Conjugates You have learned about the concept of rationalizing denominators to simplify rational expressions. You have to multiply numerator and denominator of the expression by something that is intentionally chosen to cancel out the radical. This can be problematic though, when the radical is not all by itself like, in the previous lesson. In this lesson, you will learn how to use what is called the conjugate to be able to multiply away radicals in binomial terms. The conjugate of a binomial can be found very easily by simply negating the addition or subtraction sign between the two terms. In this example, [5 over 1 minus the square root of 3] where you wish to be rid of the exponent in the denominator, you would need to multiply by the conjugate of the one minus the square root of three which is one plus the square root of three. If you multiply this to the numerator and denominator of the fraction you will not have changed the fraction's value but when you distribute out and combine like terms you will find that the denominator no longer has a square root in it. [5 over 1 minus the square root of 3 times (1 plus the square root of 3) over (1 plus the square root of 3). 5 plus 5 square root of 3 over 1 plus square root of 3 minus square root of 3 minus 3. 5 plus 5 square root of 3 over negative 2] By convention, you don't usually leave the negative in the denominator in these problems so you would multiply numerator and denominator by negative one to get the final simplified answer. [Negative 5 minus 5 square root of 3 over 2] [2 plus the square root of 5 over 6 minus the square root of 3] On this problem you will need to be a little more careful with your multiplication but there is very little that is different. Your primary goal should still be to get rid of the radical in the denominator and you will need to use the conjugate to do so. [(2 plus the square root of 5) over (6 minus the square root of 3) times (6 plus the square root of 3) over (6 plus the square root of 3)] In multiplying by the conjugate you see that the multiplication in the numerator is going to require a little more attention. [12 plus 2 square root of 3 plus 6 square root of 5 plus square root of 15 over 36 plus 6 square root of 3 minus 6 square root of 3 minus 3] Remember that multiplying the square root of three by the square root of five would be the square root of 15. Otherwise when you add the terms you will need to consolidate terms where possible but two square root of three and six square root of five cannot be consolidated in any way and must be left [12 plus 2 square root of 3 plus 6 square root of 5 plus square root of 15 over 33.

fractional exponent

an exponent in the form of a fraction, with the numerator representing the power to which the base is to be raised and the denominator representing the index of the radical

law of radicals are rules about how to combine and simplify radical expressions. Objectives

change a radical expression to the equivalent expression with fractional exponents and (Evaluate and simplify radical expressions and fractional exponent expressions)

what are the steps to simplify?

rationalize the denominator, multiply and combine terms, simplify to lowest terms

conjugate

the binomial radical expression that differs from the give expression only in the arithmetic sign between the terms ( also you can change the sign like this example The conjugate of 4 square root of 9 plus 6 is 4 square root of 9 minus 6. The difference is plus 6 and minus 6.)

what happens when the denominator of a fraction is a binomial

the fraction can be rationalized by multiplying both the numerator and the denominator of the fraction by the conjugate of the denominator

how do you rationalize a second degree binomial radical denominator?

the numerator and denominator of the fraction must both be multiplied by the conjugate of the denominator, any number or term divided by itself is equal to one therefore multiplying both the numerator and denominator by the conjugate will not change the value of the original.

rationalize

to remove radicals in the denominator by changing the fraction to an equal expression with a rational denominator

How many terms does a binomial have?

two