Use the properties of exponents and radicals to generate an equivalent expression for ((root(256, 10)) ^ 5/512 1/3
To simplify the expression ((root(256, 10))^5)/(512)^{(1/3)}, let's simplify each component separately.
First, we need to find the value of the radicand root(256, 10).
Since 256 is a perfect fourth power of 4 (4^4 = 256), we can rewrite root(256, 10) as the fourth root of 10 multiplied by the fourth root of 10 multiplied by the fourth root of 10 multiplied by the fourth root of 10.
So, root(256, 10) = (10^(1/4))*(10^(1/4))*(10^(1/4))*(10^(1/4)) = 10^(1/4 + 1/4 + 1/4 + 1/4) = 10^(4/4) = 10^1 = 10.
Now let's simplify (512)^(1/3):
The cube root of 512 is 8 because 8^3 = 512.
Now, we substitute these simplifications back into our original expression:
((root(256, 10))^5)/(512)^(1/3) = (10^5)/(8) = 100,000/8 = 12,500.
Therefore, ((root(256, 10))^5)/(512)^(1/3) is equivalent to 12,500.
Are you sure
Apologies for the mistake. Let's go through the solution again:
We want to simplify the expression ((root(256, 10))^5)/(512)^(1/3).
First, let's simplify the radicand root(256, 10):
Since 256 is a perfect 8th power of 2 (2^8 = 256), we can rewrite root(256, 10) as the eighth root of 10.
So, root(256, 10) = 10^(1/8).
Now let's simplify (512)^(1/3):
The cube root of 512 is 8 because 8^3 = 512.
Substituting these simplifications back into our original expression:
((root(256, 10))^5)/(512)^(1/3) = (10^(1/8))^5/8 = 10^(5/8)/8.
Therefore, ((root(256, 10))^5)/(512)^(1/3) simplifies to 10^(5/8)/8.
Note: This expression cannot be simplified further as the bases and exponents are different.