Algebraic Identities

If $\frac{l}{\sqrt{10}+\sqrt{14}+\sqrt{15}+\sqrt{21}}=\frac{{\displaystyle \sqrt{10}-\sqrt{14}-\sqrt{15}+\sqrt{21}}}{{\displaystyle k}}$, then

If $x^{\frac{1}{3}}+y^{\frac{1}{3}}+z^{\frac{1}{3}}=0$, then what is ${\left(x+y+z\right)}^3$ is equal to

If $b=\sqrt{\sqrt{6}-\sqrt{2}}$ and $h=\sqrt{\sqrt{6}+\sqrt{2}}$, then find the value of $\frac{1}{2}\times b\times h$.

If $x=p^{\frac{1}{3}}+p^{-\frac{1}{3}}$ and $y=p^{\frac{1}{3}}-p^{-\frac{1}{3}}$, then find the value of $\frac{(x^2-y^2{)}^3}{(x^2+y^2+xy)(x^2+y^2-xy)}$.

If $a^3+\frac{1}{a^3}=52$, then the value of $2(a+\frac{1}{a})$ is:

Find: $(3+\sqrt{3})(3-\sqrt{3})$.

$8a^3-125b^3$ when divided by $4a^2+10ab+25b^2$ gives 

Using the identity $a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)$, factorise:

$128x^3+54$

Using the identity $a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)$, factorise:

$27y^3+125z^3$

Try to write the value of $\frac{x^6-1}{x^3}$ if $\frac{x^2-1}{x}=2$.

${\left(a-b\right)}^3=$

Expand the expression $(1-3x{)}^3$

Simplify $\frac{x^3}{x-y}+\frac{y^3}{y-x}$

On simplification $(3+\sqrt{3})(3-\sqrt{3})$ gives:

Simplify: $(y+1{)}^3+(y-1{)}^3$

Simplify:

 ${\left(7x+y\right)}^3-{\left(7x+y\right)}^2$

If $p+q+r= 0$, then $\frac{p^2}{qr}+\frac{q^2}{pr}+\frac{r^2}{pq}= ?$

Expand: $x^3+y^3+z^3-3xyz$

What is $a^2+b^2$

If $x=2+\sqrt{3}$ and $y=2-\sqrt{3}$, find the value of $x^{-3}+y^{-3}$