1. Expand and simplify
a) (2-x)^-2
b) (2+x)^-2
c) General formula for
i) (x+y)^-n
ii) (x-y)^-n
iii) (x+y)^1/n
iv) (x+y)^n/m
v) (x+y)^-1/n
vi) (x+y)^-n/m
vii) (x-y)^1/n
viii) (x-y)^n/m
ix) (x-y)^-1/n
x) (x-y)^-n/m
Using Binomial theory and combination respectively
2. Write the Paschal Triangle for
a) (2-x)^-2
b) (2+x)^-2
c) General formula for
i) (x+y)^-n
ii) (x-y)^-n
iii) (x+y)^1/n
iv) (x+y)^n/m
v) (x+y)^-1/n
vi) (x+y)^-n/m
vii) (x-y)^1/n
viii) (x-y)^n/m
ix) (x-y)^-1/n
x) (x-y)^-n/m?
Where did you get stuck?
all of these just use the binomial theorem:
(a+b)^n = nC0 a^n b^0 + nC1 a^(n-1) b^1 + ... + nCk a^(n-k) b^k + ...
To expand and simplify the given expressions and derive the general formulas using binomial theorem and combination, we'll need to use the following concepts:
1. Binomial Theorem: The binomial theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a positive integer. The formula is as follows:
(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n
Here, C(n, r) represents the combination of choosing r elements from n elements, which can be calculated using the formula: C(n, r) = n! / (r!(n-r)!).
2. Negative Exponents: To simplify expressions with negative exponents, we'll need to apply the rules of exponents. Specifically, a^(-n) = 1 / a^n.
Now, let's proceed with expanding and simplifying the given expressions and deriving the general formulas:
1a) (2-x)^-2:
Using the negative exponent rule, we can rewrite (2-x)^-2 as 1 / (2-x)^2.
Expanding the expression (2-x)^2 using the binomial theorem:
(2-x)^2 = C(2, 0) * 2^2 * (-x)^0 + C(2, 1) * 2^1 * (-x)^1 + C(2, 2) * 2^0 * (-x)^2
= 1 * 4 * 1 + 2 * 2 * (-x) + 1 * 1 * x^2
= 4 - 4x + x^2
Therefore, (2-x)^-2 = 1 / (4 - 4x + x^2).
1b) (2+x)^-2:
Similarly, using the negative exponent rule, we rewrite (2+x)^-2 as 1 / (2+x)^2.
Expanding the expression (2+x)^2 using the binomial theorem:
(2+x)^2 = C(2, 0) * 2^2 * x^0 + C(2, 1) * 2^1 * x^1 + C(2, 2) * 2^0 * x^2
= 1 * 4 * 1 + 2 * 2 * x + 1 * 1 * x^2
= 4 + 4x + x^2
Therefore, (2+x)^-2 = 1 / (4 + 4x + x^2).
1c) General formulas:
i) (x+y)^-n:
Using the negative exponent rule, (x+y)^-n can be rewritten as 1 / (x+y)^n.
ii) (x-y)^-n:
Using the negative exponent rule, (x-y)^-n can be rewritten as 1 / (x-y)^n.
iii) (x+y)^1/n:
Using the nth root rule, (x+y)^1/n can be rewritten as the nth root of (x+y).
iv) (x+y)^n/m:
Using the exponentiation rule, (x+y)^n/m can be expanded as ((x+y)^n)^(1/m).
v) (x+y)^-1/n:
Using the negative exponent rule, (x+y)^-1/n can be rewritten as 1 / ((x+y)^1/n).
vi) (x+y)^-n/m:
Using the negative exponent rule, (x+y)^-n/m can be rewritten as 1 / ((x+y)^n/m).
vii) (x-y)^1/n:
Using the nth root rule, (x-y)^1/n can be rewritten as the nth root of (x-y).
viii) (x-y)^n/m:
Using the exponentiation rule, (x-y)^n/m can be expanded as ((x-y)^n)^(1/m).
ix) (x-y)^-1/n:
Using the negative exponent rule, (x-y)^-1/n can be rewritten as 1 / ((x-y)^1/n).
x) (x-y)^-n/m:
Using the negative exponent rule, (x-y)^-n/m can be rewritten as 1 / ((x-y)^n/m).
2. Pascal's Triangle:
To write Pascal's Triangle for the given expressions, we'll utilize the coefficients obtained from the binomial expansion.
a) (2-x)^-2:
The coefficients in the expansion (2-x)^-2 = 1 / (4 - 4x + x^2) represents the elements of Pascal's Triangle for the corresponding powers of (2-x).
First row: 1
Second row: 1, -4, 4
Third row: 1, -8, 14, -8, 1
And so on.
b) (2+x)^-2:
Similarly, the coefficients in the expansion (2+x)^-2 = 1 / (4 + 4x + x^2) represents the elements of Pascal's Triangle for the corresponding powers of (2+x).
First row: 1
Second row: 1, 4, 4
Third row: 1, 8, 14, 8, 1
And so on.
c) General Pascal's Triangle:
The coefficients obtained from the binomial expansion for the general formulas will represent the elements of Pascal's Triangle corresponding to those expressions.
Note: The size of Pascal's Triangle will depend on the values of n and m in the general formulas.