Use rational exponents to simplify the radical. If rational exponents appear after simplifying, write the answer in radical notation.
3sqrt(2)/ 5sqrt(2)
3√2/(5√2) = 3/5
why make it difficult by going into all kinds of rational exponents?
use rational exponents to write 4sqrt(6) * 2sqrt(7) as a single radical expression
To simplify the expression (3√(2))/(5√(2)), we can rewrite the radicals using rational exponents.
The cube root (√3) can be written as (2/3)²³and the 5th root (√5) can be written as (2/5)²⁵.
Applying these rational exponents to the expression, we get:
[(2/3)²³ * √(2)] / [(2/5)²⁵ * √(2)]
Simplifying the expression further:
[(2²³ * √(2))] / [(2²⁵ * √(2))]
Now, we can simplify the exponents:
[2^(23/3) * √(2)] / [2^(25/5) * √(2)]
Applying the properties of exponents, we subtract the exponent values:
[2^(23/3 - 25/5) * √(2)]
Simplifying the exponent:
[2^((115 - 75)/15) * √(2)]
[2^(40/15) * √(2)]
The exponent 40/15 can be simplified as 8/3:
[2^(8/3) * √(2)]
Finally, we write the expression using radical notation:
∛(2^8) * √(2)
Which simplifies to:
2^(8/3) * √(2)
To simplify the given expression involving radicals using rational exponents, we can rewrite the expression using rational exponents. The square root (√) can be expressed as a fractional exponent of 1/2.
First, let's rewrite the expression as follows:
3√2 / 5√2
Now, let's convert the square roots to fractional exponents:
(2^(1/3)) / (2^(1/5))
Since both terms have the same base (2), we can use the quotient rule of exponents to simplify this expression. According to the quotient rule, when dividing two terms with the same base, we subtract the exponents:
2^(1/3 - 1/5)
Now, let's simplify the exponent in the numerator:
2^(5/15 - 3/15)
2^(2/15)
The answer, using rational exponents, is 2^(2/15).
If you prefer the answer in radical notation, we can convert the rational exponent back into a square root. For this, we raise the base (2) to the numerator (2) and take the denominator (15) as the index of the radical. Hence, the answer in radical notation is:
∛(2^2) or ∛4