%5E3.jpeg "(2c+5d)^3")
Expanding (2c + 5d)^3
The expression (2c + 5d)^3 represents the cube of the binomial (2c + 5d). To expand this expression, we can use the binomial theorem or a simpler method of repeated multiplication.
Using the Binomial Theorem
The binomial theorem states that for any positive integer n:
(a + b)^n = ∑(n choose k) a^(n-k) b^k
where k ranges from 0 to n and (n choose k) is the binomial coefficient, which can be calculated as:
(n choose k) = n! / (k! * (n-k)!)
In our case, n = 3, a = 2c, and b = 5d. Therefore, we can expand the expression as:
(2c + 5d)^3 = (3 choose 0) (2c)^3 (5d)^0 + (3 choose 1) (2c)^2 (5d)^1 + (3 choose 2) (2c)^1 (5d)^2 + (3 choose 3) (2c)^0 (5d)^3
Calculating the binomial coefficients and simplifying the terms:
(2c + 5d)^3 = 1 * 8c^3 * 1 + 3 * 4c^2 * 5d + 3 * 2c * 25d^2 + 1 * 1 * 125d^3
Finally, we get the expanded form:
(2c + 5d)^3 = 8c^3 + 60c^2d + 150cd^2 + 125d^3
Using Repeated Multiplication
We can also expand the expression by repeatedly multiplying the binomial by itself:
(2c + 5d)^3 = (2c + 5d) * (2c + 5d) * (2c + 5d)
First, expand the first two binomials:
(2c + 5d) * (2c + 5d) = 4c^2 + 10cd + 10cd + 25d^2 = 4c^2 + 20cd + 25d^2
Now, multiply this result by the remaining binomial:
(4c^2 + 20cd + 25d^2) * (2c + 5d) = 8c^3 + 40c^2d + 50cd^2 + 20c^2d + 100cd^2 + 125d^3
Combining like terms:
(2c + 5d)^3 = 8c^3 + 60c^2d + 150cd^2 + 125d^3
As you can see, both methods lead to the same result.
Therefore, the expanded form of (2c + 5d)^3 is 8c^3 + 60c^2d + 150cd^2 + 125d^3.