Answer:
Explanation:
To simplify the expression, we can combine the square roots and simplify the exponents.
Starting with the expression:
3√(16x^7) * 3√(12x^9)
Let's simplify each term separately:
Simplifying 3√(16x^7):
The index of the radical is 3, so we need to group the terms in sets of three. For the variable x, we have x^7, which can be grouped as x^6 * x.
Now, let's simplify the number inside the radical:
16 = 2^4, and we can rewrite it as (2^3) * 2 = 8 * 2.
So, 3√(16x^7) becomes:
3√(8 * 2 * x^6 * x) = 2 * x^2 * 3√(2x)
Simplifying 3√(12x^9):
Again, the index of the radical is 3, and we group the terms in sets of three. For the variable x, we have x^9, which can be grouped as x^6 * x^3.
Now, let's simplify the number inside the radical:
12 = 2^2 * 3.
So, 3√(12x^9) becomes:
3√(2^2 * 3 * x^6 * x^3) = 2 * x^2 * 3√(3x^3)
Now we can multiply the simplified terms together:
(2 * x^2 * 3√(2x)) * (2 * x^2 * 3√(3x^3))
Multiplying the coefficients: 2 * 2 * 3 = 12.
Multiplying the variables: x^2 * x^2 = x^4.
Now, let's combine the square roots:
3√(2x) * 3√(3x^3) = 3√(2x * 3x^3) = 3√(6x^4).
Therefore, the simplified expression is:
12x^4 * 3√(6x^4)