I need to rationalize the denominator of 2/3 square root of five, (three being the index). My book suggests you should multiply this by 3 square root of 5^2 over 3 square root of 5^2. I thought the numbers that you rationalize the original with should be the denominator unaltered. Why do you add a ^2 to the 5? Thanks for your help!!!
are you sure this is not
2
____
5^(1/3)
?
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If that is the case, you need to get 5^1 = 5 on the bottom
The way to do that is to multiply top and bottom by
5^(2/3)
2 * 5^(2/3)
______________
5^(1/3) * 5^(2/3)
which is
2 * 5^2/3
__________
5
To rationalize the denominator of the expression (2/3)√5^3 (with 3 being the index), you can follow the steps provided in your book:
Step 1: Multiply the expression by a form of 1 that eliminates the radical in the denominator.
In this case, your book suggests multiplying by (3√5^2) / (3√5^2).
Step 2: The reason for squaring the 5 in the denominator is to simplify it, making it easier to eliminate the radical.
Squaring the number effectively removes the radical sign. In this case, squaring √5^3 simplifies to 5^3.
Step 3: Multiply the numerator and denominator by the same expression:
(2/3)√5^3 * (3√5^2) / (3√5^2)
Step 4: Apply the property of radicals that states √(a * b) = √a * √b.
This property allows us to separate the radical expression:
(2/3)* 3√(5^3 * 5^2) / (√(5^2))^3
Step 5: Simplify the expression inside the radical:
(2/3) * 3√(5^5) / (√5^2)^3
= (2/3) * 3√(5^5) / (5^3)
Step 6: Simplify further:
(2/3) * 3 * 5^2 * √5 / 5^3
= 2 * 5^2 * √5 / 3 * 5^3
Step 7: Cancel out common factors between the numerator and denominator:
2 * 5^2 * √5 / 3 * 5^3
= 2 * 5^2/5^3 * √5
= 2 * 1/5 * √5
= 2/5 * √5
So, the rationalized denominator of (2/3)√5^3 is (2/5)√5.