How would you simplify the cubed root of 54x^17
??
(54x^17)^(1/3)
= (27x^15)^(1/3) (2x^2)^(1/3)
= 3x^5 (2^(1/3) x^(2/3)
First I factor 54.
5 4
| |
9 6
|\ \ \
3 3 3 2
{3 3 3} group of threes that can be out of cubed root
Because there are three threes I take them out of the cubed root (# = im going to consider this as my cubed root button) turning #54x^17 into 3#2x^17
Now we factor x^17:
there are 5 groups of 3s in 17 so we would take those out leaving x^2 in the root.
3#2x^17 turns into 3x^5 # 2x^2
Then this simplified is therefore
3x^5 cube root of 2x^2
To simplify the cube root of 54x^17, you can break it down into prime factors. First, let's split 54 into its prime factors:
54 = 2 x 3 x 3 x 3
Next, simplify x^17 into its prime factors:
x^17 = x^16 x x
= (x^2)^8 x x
Now we can take the cube root of 54x^17:
∛(54x^17) = ∛(2 x 3 x 3 x 3 x (x^2)^8 x x)
Since we are looking for the cube root, we can simplify it further by grouping the prime factors to have three of a kind:
∛(2 x 3 x 3 x 3 x (x^2)^8 x x) = ∛(2 x (3^3) x ((x^2)^2)^4 x x)
= ∛(2 x 3^3 x x^2 x x^2 x x^2 x x^2 x x)
= ∛(2 x 3^3 x x^2 x x^2 x x^2 x x^3)
= 3x^6 ∛(2x^2)
Therefore, the simplified form of the cube root of 54x^17 is 3x^6 ∛(2x^2).
To simplify the cube root of 54x^17, we can break it down into two parts:
Step 1: Simplify the number part (54)
To simplify the cube root of a number, we need to find its prime factors. The prime factorization of 54 is 2 * 3^3.
Step 2: Simplify the variable part (x^17)
To simplify the cube root of a variable expression, we can divide the exponent by 3 and take the remaining exponent. In this case, we have x^17. Since 17 divided by 3 is 5 with a remaining exponent of 2, we have x^5.
Combining the simplified number part and variable part, we get:
Cube root of 54x^17 = Cube root of (2 * 3^3 * x^17)
= Cube root of (2 * 3 * 3 * 3 * x^5)
= 3 * x^5 * Cube root of 2
Therefore, the simplified form of the cube root of 54x^17 is 3x^5 * Cube root of 2.