Sqrt(18x^27)
To simplify the square root of 18x^27, we can break it down using the properties of exponents.
First, we can rewrite 18x^27 as the product of two terms: sqrt(18) * sqrt(x^27).
Then, we can simplify each term separately:
1. sqrt(18):
Since 18 can be factored into 9 * 2 = 3^2 * 2, we have sqrt(18) = sqrt(3^2 * 2) = sqrt(3^2) * sqrt(2) = 3 * sqrt(2).
2. sqrt(x^27):
We can rewrite x^27 as (x^13)^2 * x:
sqrt(x^27) = sqrt((x^13)^2 * x) = x^13 * sqrt(x).
Combining the simplified terms, we have:
sqrt(18x^27) = sqrt(18) * sqrt(x^27) = 3 * sqrt(2) * x^13 * sqrt(x) = 3x^13 * sqrt(2x).
To simplify the expression sqrt(18x^27), we can break it down into its factors and simplify each part separately.
Step 1: Start by breaking down 18x^27 into its prime factors:
18 = 2 * 3^2
x^27 = x^(3 * 9) = (x^3)^9
Step 2: Rewrite sqrt(18x^27) using the prime factors:
sqrt(18x^27) = sqrt(2 * 3^2 * (x^3)^9)
Step 3: Simplify the square root of each factor:
sqrt(2) * sqrt(3^2) * sqrt((x^3)^9)
Step 4: Simplify each square root:
sqrt(2) * 3 * x^(3 * 9/2)
Step 5: Simplify the exponent:
sqrt(2) * 3 * x^13.5
Step 6: Since the exponent is a fractional value, we can rewrite it using the root notation:
sqrt(2) * 3 * (x^(1/2))^27
So, the simplified expression for sqrt(18x^27) is 3(sqrt(2)) * x^13.5