Question

[(4+√15)^3/2] -[(4-√15)^3/2]=k√6​

Answer 1

The value of k is 9.

Step-by-step explanation:

Given : Expression $[(4+\sqrt{15})^{\frac{3}{2}}]-[(4-\sqrt{15})^{\frac{3}{2}}]=k\sqrt6$

To find : The value of k ?

Solution :

Re write the expression as,

$[(4+\sqrt{15})^{\frac{1}{2}}]^3-[(4-\sqrt{15})^{\frac{1}{2}}]^3=k\sqrt6$

Applying algebraic identity, $a^3-b^3=(a-b)(a^2+b^2+ab)$

Here, a=(4+\sqrt{15})^{\frac{1}{2}} and b=(4-\sqrt{15})^{\frac{1}{2}}

Substitute,

$((4+\sqrt{15})^{\frac{1}{2}}-(4-\sqrt{15})^{\frac{1}{2}})[(4+\sqrt{15})^{\frac{2}{2}}+(4-\sqrt{15})^{\frac{2}{2}}+((4+\sqrt{15})^{\frac{1}{2}})((4-\sqrt{15})^{\frac{1}{2}})]=k\sqrt6$

$((4+\sqrt{15})^{\frac{1}{2}}-(4-\sqrt{15})^{\frac{1}{2}})[8+(16-15)^{\frac{1}{2}}]=k\sqrt6$

$((4+\sqrt{15})^{\frac{1}{2}}-(4-\sqrt{15})^{\frac{1}{2}})[9]=k\sqrt6$

Squaring both side,

$81[4+\sqrt{15}+4-\sqrt{15}-2[(4+\sqrt{15})(4-\sqrt{15})]^{\frac{1}{2}}]=k^2(\sqrt6)^2$

$81[8-2]=6k^2$

$81\times 6=6k^2$

$k^2=81$

$k=9$

Therefore, the value of k is 9.

#Learn more

Simplify (4+root 7) (3+root 2)

https://brainly.in/question/9469746