Verify the Identity x^3+y^3=(x+y)(x^2-xy+y^2)

$x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})$

Simplify the right side.

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Simplify $(x + y) (x^{2} - x y + y^{2})$ .

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Expand $(x + y) (x^{2} - x y + y^{2})$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} + y^{3} = x \cdot x^{2} + x (- x y) + x y^{2} + y x^{2} + y (- x y) + y \cdot y^{2}$

Simplify terms.

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Simplify each term.

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Multiply $x$ by $x^{2}$ by adding the exponents.

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Multiply $x$ by $x^{2}$ .

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Raise $x$ to the power of $1$ .

$x^{3} + y^{3} = x^{1} x^{2} + x (- x y) + x y^{2} + y x^{2} + y (- x y) + y \cdot y^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} + y^{3} = x^{1 + 2} + x (- x y) + x y^{2} + y x^{2} + y (- x y) + y \cdot y^{2}$

Add $1$ and $2$ .

$x^{3} + y^{3} = x^{3} + x (- x y) + x y^{2} + y x^{2} + y (- x y) + y \cdot y^{2}$

Rewrite using the commutative property of multiplication.

$x^{3} + y^{3} = x^{3} - x (x y) + x y^{2} + y x^{2} + y (- x y) + y \cdot y^{2}$

Multiply $x$ by $x$ by adding the exponents.

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Move $x$ .

$x^{3} + y^{3} = x^{3} - (x \cdot x) y + x y^{2} + y x^{2} + y (- x y) + y \cdot y^{2}$

Multiply $x$ by $x$ .

$x^{3} + y^{3} = x^{3} - x^{2} y + x y^{2} + y x^{2} + y (- x y) + y \cdot y^{2}$

Rewrite using the commutative property of multiplication.

$x^{3} + y^{3} = x^{3} - x^{2} y + x y^{2} + y x^{2} - y (x y) + y \cdot y^{2}$

Multiply $y$ by $y$ by adding the exponents.

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Move $y$ .

$x^{3} + y^{3} = x^{3} - x^{2} y + x y^{2} + y x^{2} - (y \cdot y) x + y \cdot y^{2}$

Multiply $y$ by $y$ .

$x^{3} + y^{3} = x^{3} - x^{2} y + x y^{2} + y x^{2} - y^{2} x + y \cdot y^{2}$

Multiply $y$ by $y^{2}$ by adding the exponents.

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Multiply $y$ by $y^{2}$ .

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Raise $y$ to the power of $1$ .

$x^{3} + y^{3} = x^{3} - x^{2} y + x y^{2} + y x^{2} - y^{2} x + y^{1} y^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} + y^{3} = x^{3} - x^{2} y + x y^{2} + y x^{2} - y^{2} x + y^{1 + 2}$

Add $1$ and $2$ .

$x^{3} + y^{3} = x^{3} - x^{2} y + x y^{2} + y x^{2} - y^{2} x + y^{3}$

Combine the opposite terms in $x^{3} - x^{2} y + x y^{2} + y x^{2} - y^{2} x + y^{3}$ .

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Reorder the factors in the terms $- x^{2} y$ and $y x^{2}$ .

$x^{3} + y^{3} = x^{3} - x^{2} y + x y^{2} + x^{2} y - y^{2} x + y^{3}$

Add $- x^{2} y$ and $x^{2} y$ .

$x^{3} + y^{3} = x^{3} + x y^{2} + 0 - y^{2} x + y^{3}$

Add $x^{3} + x y^{2}$ and $0$ .

$x^{3} + y^{3} = x^{3} + x y^{2} - y^{2} x + y^{3}$

Reorder the factors in the terms $x y^{2}$ and $- y^{2} x$ .

$x^{3} + y^{3} = x^{3} + y^{2} x - y^{2} x + y^{3}$

Subtract $y^{2} x$ from $y^{2} x$ .

$x^{3} + y^{3} = x^{3} + 0 + y^{3}$

Add $x^{3}$ and $0$ .

$x^{3} + y^{3} = x^{3} + y^{3}$

Since the two sides have been shown to be equivalent, the equation is an identity.

$x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})$ is an identity.

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Name

Name	one billion eighty-eight million nine hundred fifty-seven thousand forty-eight

Interesting facts

1088957048 has 64 divisors, whose sum is 3777254208
The reverse of 1088957048 is 8407598801
Previous prime number is 313

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 50
Digital Root 5

Name

Name	one billion eight hundred forty-two million seven hundred seventeen thousand eight hundred three

Interesting facts

1842717803 has 4 divisors, whose sum is 1842811968
The reverse of 1842717803 is 3087172481
Previous prime number is 66421

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 41
Digital Root 5

Name

Name	three hundred three million seven hundred seven thousand ninety-three

Interesting facts

303707093 has 8 divisors, whose sum is 313904640
The reverse of 303707093 is 390707303
Previous prime number is 839

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 32
Digital Root 5

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