^5 sqrt symbol a^6/ ^4 sqrt symbol a^3
To simplify the expression ^5√(a^6) / ^4√(a^3), we can start by simplifying the individual roots separately.
First, let's simplify the numerator, ^5√(a^6). Since the index of the root is 5, we need to find a number that when raised to the power of 5 equals a^6. In this case, it's a^6 itself since (a^6)^(1/5) = a^(6/5). So ^5√(a^6) simplifies to a^(6/5).
Next, let's simplify the denominator, ^4√(a^3). The index of the root is 4, so we need to find a number that when raised to the power of 4 equals a^3. In this case, it's a^(3/4) since (a^3)^(1/4) = a^(3/4). So ^4√(a^3) simplifies to a^(3/4).
Now we can rewrite the expression as a^(6/5) / a^(3/4). To divide two numbers with the same base, we subtract their exponents. So a^(6/5) / a^(3/4) equals a^(6/5 - 3/4).
To simplify the exponent, we need to find a common denominator for 5 and 4, which is 20. Now we rewrite the expression as a^(24/20) / a^(15/20).
Again, to divide two numbers with the same base, we subtract their exponents. So a^(24/20) / a^(15/20) equals a^(24/20 - 15/20).
Subtracting the exponents, we have a^(9/20).
Therefore, the simplified expression is ^9√a^20.