Section 5 7: Rational Exponents Simplify each of the following expressions to the furthest extent possible. You should have gotten 2xy 4 for the first one, 2x 2 y 3 for the second one, and concluded that the third question is impossible. That last problem appears to be impossible to do because each expression does not have the same index Kind of like like terms in variable expressions, like radical expressions (radicals with the same indices and radicands) can be added and subtracted with each othe only if they are like each other. But, kind of like unlik terms in variable expressions, unlike radical expressions can still be multiplied (and divided) by each other. radicals are just exponents What number, multiplied by itself THREE times, yields 11? What number, multiplied by itself THREE times, yields 11? Solution: So if both of these statements can be said to make 11 when multiplied by themselves three times, what must be true? 1
For all radical expressions, b p/n. NOTE: Some of those fraction exponents will be reducible... Note, too, how nice it is to think of the index as being a denominator... It makes problems like possible to deal with. Even if we can't come up with a perfect little rational value to place that expression, we can still simplify or sort of 'reduce' it... 1.) 3.) 2.) 4.) The reason that we can multiply radical expressions that appear to be unlike each other (because they have different indices) is that we can convert the radical expressions into rational exponent expressions, get a common denominator for the fraction exponents, and then go ahead and multiply the values by adding the fractional exponents. The final answers should have either fully simplified fraction exponents, or else be similarly reduced and put back into radical form. 2
Now we'll get to simplify problems that just didn't look like they could be simplified before. Remember the Product of Powers Rule (when you are multiplying powers with the same base, you keep the base and add the exponents!) You are adding fractions here, remember to get a common denominator!! To do problems like those shown below, first, rewrite each radicand as a prime number raised to a power (if it's not like that already). Then, turn each expression into its rational exponent form. Next, make the fraction exponents have a common denominator, too, so that you'll be able to add the powers. Finally, you can rewrite the base (former radicand) and actually add the exponents. Reduce, if possible. Rewrite in radical form (answers should be written in the same format as the question). 1.) 2.) Add Reduce 3.) 4.) 5.) 6.) Just rewrite the base of the exponential expression as a prime number raised to a power. Section 5 7, Day 2 Using this concept of rewriting values using rational exponents is also something we do to evaluate or simplify some individual exponential expressions, such as 125 2/3 or 16 3/4. Put that value in parentheses and continue the problem by multiplying the exponent on the rewritten base with the exponent that was originally on the value to simplify. For example: 1.) 125 2/3 2.) 16 3/4 3.) 64 1/6 4.) 64 5/3 3
You could also be asked to do problems like: 1.) 25 1/2 * 25 3/2 2.) 8 2/3 * 8 1/3 3.) 4 1/2 * 16 3/2 4.) 4 1/2 * 8 2/3...Day 3...? (After we go back into 5 6, too.) 5.)x 2/3 * x 4/3 6.) y 3/4 * y 1/4 7.) (h 2/5 ) 3 8.) (g 1/3 ) 3/5 QUICK CHECK OF YOUR UNDERSTANDING 9.) 10.) RATIONALIZING DENOMINATORS RATIONALIZING THE DENOMINATOR a.k.a. getting radicals out of the basement... Basically, a value is not considered to be fully simplified if it has radicals in its denominator. In order to fix a rule breaker, we just multiply that fraction by a form of one (just like we do when we want to change a fraction s denominator to make a common denominator). When the goal is to rationalize the denominator, we multiply the top & bottom of the fraction by something that will turn the denominator into a rational number. If it was a single part denominator (like we just saw & like we're about to see), we multiply in a value (using the same kind of radical) that will make the magic number we need to make the radicand (quantity under the radical sign) a perfect square, cube, or whatever type of root it is. For instance, the square root of "3" would be rational square if it were the square root of "3 times 3". So, we bring in another 3! 4
Here are some more challenging ones... And keep in mind situations like x 1/3 or 5y 3/4 and what they really mean (and therefore what they require us to fix)... x 1/3 = 5y 3/4 = End of plan for today. It's a pretty different story, though, when you are confronted with denominators like these. Any ideas? The way we fix things if there is a binomial like radical expression in the denominator is to multiply by the form of one (top & bottom) that uses the conjugate of the pesky denominator ( a + b or a b, as needed). Then, we simplify the top and bottom to the fullest extent possible....for example: check yo'self before you wreck yo'self! 1.) 2.) 3.) 4.) 5
check yo'self don't wreck yo'self...math camp buddy, Sam Shah... What a character! THE EXPONENT RULES WITH NON INTEGERS! Simplify: check yo'self before you wreck yo'self 6