Find the cube root of 27a3+54ab2+36a2b+8a3
I think you mean
27a^3+54ab^2+36a^2b+8b^3
Note that 27a^3 and 8b^3 are perfect cubes
So the first thing that comes to mind is
(3a+2b)^3
I think you will find that that is what you want.
Well, well, well. It seems like you want to find the cube root of a rather complex expression. Don't worry, I'm here to untangle this mathematical web for you!
Let's break it down, shall we?
First, we have 27a^3, which is a "cube" expression. So, the cube root of that would simply be 3a.
Next, we have 54ab^2. This term doesn't have a perfect cube within it, so we'll just have to leave it as is.
Moving on, we encounter 36a^2b. Again, no perfect cube here, so it remains unchanged.
Finally, we have 8a^3. Ah, another cube! So the cube root of that would be 2a.
Now let's put it all together!
The cube root of 27a^3 + 54ab^2 + 36a^2b + 8a^3 is:
(3a + 2a) + 54ab^2 + 36a^2b
= 5a + 54ab^2 + 36a^2b
And there you have it! The cube root of the given expression is 5a + 54ab^2 + 36a^2b.
To find the cube root of the expression 27a^3 + 54ab^2 + 36a^2b + 8a^3, we can break it down and simplify it step by step:
Step 1: Group the terms that have the same base and exponent.
27a^3 + 8a^3 = (27 + 8)a^3 = 35a^3
54ab^2 + 36a^2b = 54ab(b) + 36ab(a) = 90ab(b + a)
Now the expression becomes:
35a^3 + 90ab(b + a)
Step 2: Factor out common terms.
35a^3 + 90ab(b + a) = 5a(7a^2) + 10ab(9b + 9a)
Now the expression becomes:
5a(7a^2 + 10ab(9b + 9a))
Step 3: Simplify further if possible.
There are no more common terms to factor out, so we have reached the most simplified form of the expression.
Therefore, the cube root of 27a^3 + 54ab^2 + 36a^2b + 8a^3 is:
∛(5a(7a^2 + 10ab(9b + 9a)))
To find the cube root of the expression 27a^3 + 54ab^2 + 36a^2b + 8a^3, we can factor out the common factors from each term inside the expression.
Step 1: Group the terms with similar variables together.
27a^3 + 8a^3 + 54ab^2 + 36a^2b
Step 2: Factor out the common factor from each group.
a^3 is a common factor in the first two terms, and a is a common factor in the last two terms.
(a^3(27 + 8) + ab^2(54) + a^2b(36))
Step 3: Simplify the expression inside the parentheses.
(35a^3 + 54ab^2 + 36a^2b)
Step 4: Now we take the cube root of each term.
(cube root of 35a^3 + cube root of 54ab^2 + cube root of 36a^2b)
The cube root of 35a^3 simplifies to 5a.
The cube root of 54ab^2 simplifies to 3b√2a.
The cube root of 36a^2b simplifies to 3a√b.
So the cube root of the expression 27a^3 + 54ab^2 + 36a^2b + 8a^3 is (5a + 3b√2a + 3a√b).