Question

Find the value of 8x^3-y^3 if 2x-y=4
and xy= -1

Answer 1

Answer:

64

Explanation:

As per the information provided in the question, We have :

• 2x-y=4
• xy= -1

We are asked to find the value of 8x³ - y³.

In order to find the value of 8x³ - y³, We will simplify the given equations.

Simplifying the first equation,

$\begin{gathered}\longmapsto \rm 2x - y = 4\end{gathered}$

On cubing both sides,

$\begin{gathered} \longmapsto \rm {(2x - y)}^{3}= {(4)}^{3}\end{gathered}$

Using the identity,

⟶ A³ - b³ = (a - b)³ + 3ab(a - b)

⟶ (2x - y)³ + 3(2xy)(2x - y)

⟶ (4)³ + 3(2(-1))(4)

⟶ 64 + 3(-2)(4)

⟶ 64 + (-24)

⟶ 40

∴ Hence, 8x³ - y³ = 40.

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⠀⠀⠀⠀⠀★ Algebraic Identities :

• ( a - b )² = a² + b² - 2ab

• ( a + b )² + ( a - b)² = 2a² + 2b²

• ( a + b )² - ( a - b)² = 4ab

• ( a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ca

• a² + b² = ( a + b)² - 2ab

• (a + b )³ = a³ + b³ + 3ab ( a + b)

• ( a - b)³ = a³ - b³ - 3ab ( a - b)

• If a + b + c = 0 then a³ + b³ + c³ = 3abc

Answer 2

Question :

Find the value of 8x³–y³ if 2x-y=4 and xy= -1

Formula Used :

a³ – b³ = (a – b)³ + 3ab(a – b)

Solution :

According to the given formula we need to put the values there and simplify and get the result

$\small \sf \dashrightarrow8 {x}^{3} - {y}^{3} = (2x - y) ^{3} + 3(2x)(y)(2x - y)$

Now putting the values as according to the given condition we get

$\small \sf \dashrightarrow8 {x}^{3} - {y}^{3} = {(4)}^{3} + 3(2 \times ( - 1))(4)$

Here we have put 2 because in the first line it was given 2xy and the value of xy is -1 so 2xy means 2×(-1)

$\small \sf \dashrightarrow8 {x}^{3} - {y}^{3} = 64 + 3 \times ( - 2)(4)$

$\small \sf \dashrightarrow8 {x}^{3} - {y}^{3} = 64 - 6 \times 4$

$\small \sf \dashrightarrow8 {x}^{3} - {y}^{3} = 64 - 24$

$\small \sf \dashrightarrow8 {x}^{3} - {y}^{3} = 40$

So the value of 8x³ - y³ is 40

More Information :

• ( a + b )² = a² + b² + 2ab

• ( a - b )² = a² + b² - 2ab

• ( a + b )² + ( a - b)² = 2(a² + b²)

• ( a + b )² - ( a - b)² = 4ab

• a² + b² = ( a + b)² - 2ab

• a² + b² = ( a - b)² + 2ab

• ( a + b + c )² = a² + b² + c² + 2(ab + bc + ca)

• ( a + b - c )² = a² + b² + c² - 2(ab - bc - ca)

• (a + b )³ = a³ + b³ + 3ab ( a + b)

• ( a - b)³ = a³ - b³ - 3ab ( a - b)