Write ^4 square root k^9/^5 square root k^7 as a single radical using the smallest possible root
To simplify the expression ^4 square root k^9/^5 square root k^7, we can write it as (^4 square root k^9)/(^5 square root k^7) and simplify each radical separately.
First, let's simplify ^4 square root k^9. Since the index is 4, we can rewrite it as ^4 square root (k^9)^4. This is equivalent to ^4 square root k^36.
Next, let's simplify ^5 square root k^7. Since the index is 5, we can rewrite it as ^5 square root (k^7)^5. This is equivalent to ^5 square root k^35.
Now, we can put both of these simplified radicals back together by dividing the exponents. Therefore, ^4 square root k^9/^5 square root k^7 simplifies to ^4 square root k^(36/35).
Since the smallest possible root that contains both 4 and 35 as factors is 140, we can rewrite the expression as ^140 square root k^(36/35).