Solve Power of Sums

In mathematics, “power of sums” is come under the topic algebra. Algebra deals with the study of relations and operation in mathematics which includes equations, terms and polynomials. One of the most pure forms of mathematic is algebra. Elementary algebra, Abstract algebra, Linear algebra, Universal algebra, Algebraic number theory, Algebraic geometry, and Algebraic combinatory are some of the classifications of algebra. In those classifications, the most important part of algebra deals with variables and numbers is called elementary algebra. “Power of sums” is come under the topic elementary algebra.


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Let as assume a and b are the variables. Then, “power of sums” is given as,

(a + b) n

Where, n = 2, 3, 4 …
Standard Formulas for “power of Sums”:

For n = 2, “power of sums” equation given as,

(a + b) 2 = a 2 + 2ab+ b2

For n = 3, “power of sums” equation given as,

(a + b) 3 = a 3 + 3a2 b + 3a b2 + b3

For n = 4, “power of sums” equation given as,

(a + b) 4 = a 4 + 4a3 b + 6a2 b2 + 4a b3 + b4
Examples:

Example1: Solve the given terms: (3 + 6) 2 and (2 + 5) 2

Solution:

Given that, (3 + 6) 2 and (2 + 5) 2

For n = 2, we have using this equation,

(a + b) 2 = a 2 + 2ab+ b2

(3 + 6) 2 = 3 2 + 2*3*6+ 62

= 9 + 36 + 36

= 81

(3 + 6) 2 = 81.

(2 + 5) 2 = 2 2 + 2*2*5+ 52

= 4 + 20 + 25

= 49

(2 + 5) 2 = 49.

Example 2: Solve the given terms and prove it with normal solving?

(3 +5)3

Solution:

For n = 3, “power of sums” equation given as,

(a + b) 3 = a 3 + 3a2 b + 3a b2 + b3

(3 + 5) 3 = 3 3 + 3*32 5 + 3*3 52 + 53

= 9 + 135 + 225 + 125

= 512. ----------- (1)

Proof:

(3 + 5) 3 = (8) 3

= 8*8*8

= 512. ----------- (2)

From (1) and (2), it is proved

Example 3: solve the given terms, (4 + 7) 4 and prove it?

Solution:

Given that, (4 + 7) 4

For n = 4, “power of sums” equation given as,

(4 + 7) 4 = 44 + 4*43 7 + 6*42 72 + 4*4* 73 + 74

= 256 + 1792 + 4704 + 5488 + 2401.

= 14641. ----------- (1)

Proof:

(4 + 7) 4 = 114

= 11*11*11*11.

= 14641. ----------- (2)

From (1) and (2), it is proved.

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Example 4: Solve the given terms: (2x + 3y) 2 and (3x + 4y) 2

Solution:

Given that, (2x + 3y) 2 and (3x + 4y) 2

For n = 2, we have using this equation,

(a + b) 2 = a 2 + 2ab+ b2

(2x + 3y) 2 = 2x 2 + 2*2x*3y+ 3y2

= 4 x 2+ 12xy + 9 y2

(2x + 3y) 2 = 4x 2 + 12xy + 9y2.

(3x + 4y) 2 = 3x 2 + 2*3x*4y+ 4y2

= 9x 2+ 24xy + 16y2

(3x + 4y) 2 = 9x 2+ 24xy + 16y2

Practice problem:

Problem 1: Solve the given terms: (14 + 5) 2 and (9 + 3) 2

Answer is 361 and 144.

Problem 2: Solve the given terms: (5x + 2y) 2 and (9x + 6y) 2

Answer is 25x 2 + 20xy + 4y2.

81x 2 + 144xy + 36y2.

Problem 3: solve the given terms, (4 + 7) 4 and prove it?