What is ^4 square root 1280r^8 t^12 in simplified form?
Whoa, that's a math riddle that even I can't clown around with! The simplified form of ^4 √(1280r^8 t^12) is 4r^2 t^3 √(5). Keep rocking those math problems!
To simplify ^4√(1280r^8t^12), you can break down the expression into its prime factors and simplify them individually.
Let's start by finding the prime factorization of 1280:
1280 = 2^7 * 5
Now, let's simplify the radical expression for r^8:
r^8 = (r^4)^2
Next, let's simplify the radical expression for t^12:
t^12 = (t^6)^2
Now, let's put everything together:
^4√(1280r^8t^12) = ^4√(2^7 * 5 * r^8 * t^12)
= ^4√(2^4 * 2^3 * 5 * r^4 * r^4 * t^6 * t^6)
= 2 * r^2 * t^3 * ^4√(5)
So, the simplified form of ^4√(1280r^8t^12) is 2r^2t^3√(5).
To simplify the expression ^4 square root 1280r^8 t^12, we can break it down into separate factors and simplify each one individually.
Step 1: Break down 1280
To simplify the term ^4 square root 1280, we need to find the largest perfect fourth power that can be extracted from it. In this case, we can rewrite 1280 as (2^7 * 5).
Step 2: Simplify each factor separately
a. Simplify the fourth root of 2^7:
To simplify ^4 square root 2^7, we divide the exponent 7 by 4: 7 ÷ 4 = 1 with a remainder of 3.
Therefore, 2^7 = 2^4 * 2^3 = 16 * 2 = 32.
b. Simplify the fourth root of 5:
We cannot simplify the fourth root of 5 any further, so it remains as ^4 square root 5.
c. Simplify r^8:
The fourth root of r^8 is r^(8/4) = r^2.
d. Simplify t^12:
The fourth root of t^12 is t^(12/4) = t^3.
Step 3: Put it all together
Combining the simplified factors, we have:
^4 square root 1280r^8 t^12 = ^4 square root (32 * 5) * r^2 * t^3 = ^4 square root 32 * ^4 square root 5 * r^2 * t^3.
So, the simplified form of ^4 square root 1280r^8 t^12 becomes: 2 * ^4 square root 5 * r^2 * t^3.