How do you simplify the radical 4sqrt(128 n^8)
128 = 2 * 64
n ^ 8 = ( n ^ 4 ) ^ 2
4 sqrt ( 128 * n ^ 8 ) =
4 sqrt[ 64 * 2 * ( n ^ 4 ) ^ 2 ]
sqrt ( 64 ) = 8
sqrt [ ( n ^ 4 ) ^ 2 ] = n ^ 4
4 sqrt[ 64 * 2 * ( n ^ 4 ) ^ 2 ] =
4 * 8 * n ^ 4 * sqrt ( 2 )=
32 n ^ 4 sqrt (2 )
2n^2 ^4squareroot 8
To simplify the radical expression 4√(128n^8), you can start by breaking down the number under the radical sign into its prime factors. Then, you can simplify the expression further by applying the properties of radicals.
Step 1: Prime factorization of 128:
The prime factors of 128 are 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2^7.
Step 2: Simplify the expression by taking out perfect squares:
Since the square root of 2^2 is 2, we can take out pairs of 2's as perfect squares. In this case, we have seven 2's inside the radical, so we can simplify as 4 * 2^n^8 * 2^(7/2).
Step 3: Simplify the expression further:
We can combine the two exponents of 2, as n^8 * 2^(7/2) is equal to 2^(2*7/2) * n^8. Simplifying the exponent, we get 2^(14/2) * n^8, which is equal to 2^7 * n^8.
Step 4: Final result:
Putting everything together, the simplified form of the original expression is 8n^8 * 2^7.
Alternatively, you can write it as 8n^8 * 128.
So, the simplified radical expression is 8n^8 * 128.