Question

if y +1/4Y = 2 then find the value of 16Y^3 + 1/4 Y^3 ?

Answer 1

Concept:

Algebra is the basic concept of all Mathematics and Physics. It helps in solving problems and solutions that very much algebraic in nature.

Given:

The following expression is given

$y + \frac{1}{4y} = 2$

Find:

The value of the expression

$16{y}^{3} + \frac{1}{4 {y}^{3} }$

Solution:

Applying the formulas of Algebra,

$({a}^{3} + \frac{1}{ {a}^{3} } )$

Simplifying we get,

$16( {y}^{3} + \frac{1}{(4 {y})^{3} } ) \\ 16((y + \frac{1}{4y} )^{3} - \frac{3y}{4y} \times (y + \frac{1}{4y} )) \\ 16( {2}^{3} - \frac{3}{4} \times 2) \\ 16(8 - \frac{3}{2} ) \\ 16 \times \frac{13}{2} \\ 13 \times 8 \\ 104$

Hence, the value of rhe expression is

$104$

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Answer 2

Concept

We know that the formula for a^3+b^3 which is as follows,

a^3+b^3=(a+b)(a^2+b^2-ab)

Therefore, we can writ for the a^3+1/b^3 as follows,

a^3+1/b^3=(a+1/b)(a^2+1/b^2-a/b)

We will use the above formula to calculate the value of the given function.

Given

It is given that y+1/4y=2

Find

We have to calculate the value of 16y^3+1/4y^3.

Solution

Since, we know that a^3+1/b^3=(a+1/b)(a^2+1/b^2-a/b), therefore taking 16 common in 16y^3+1/4y^3, we have

16(y^3+1/64y^3)=16(y^3+1/(4y)^3)

=16(y+1/4y)(y^2+1/16y^2-1/4)

Now from the given condition we have

y+1/4y=2

Squaring on both side we have

y^2+1/16y^2+1/2=4

y^2+1/16y^2=4-1/2

y^2+1/16y^2=7/2

Using this into the above equation, we have

16(y+1/4y)(y^2+1/16y^2-1/4)=16*2*(7/2-1/4)

=16*2*13/4

=104

Therefore the value of 16y^3+1/4y^3 is 104.

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