Algebraic Identities

If $a+b+c=0$, then the value of $\left(\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab}\right)$ is:

If $p=101$, then find the value of $\sqrt[3]{p\left(p^2-3p+3\right)-1}$.

If $\frac{1}{x+1}+ \frac{2}{y+2} +\frac{1009}{z+1009} =1$, then find the value of  $\frac{x}{x+1}+\frac{y}{y+2}+\frac{z}{z+1009}=$?

What should come in the place of ($?$) for the expression given below?

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

Find the value of the followings

$\left(1+\frac{1}{x}\right)\left(1+\frac{1}{x+1}\right)\left(1+\frac{1}{x+2}\right)\left(1+\frac{1}{x+3}\right)$

If $x+\frac{a}{x }=b$, then find the value of $\frac{x^2+bx+a }{b{x}^2-x^3}$

If $a+\frac{1}{a}=\sqrt{3}$, find the value of $a^3+{\left(\frac{1}{a}\right)}^3$=?

The value of $\left(1+\frac{1}{\mathrm{a}}\right)\left(1+\frac{1}{\mathrm{a}+1}\right)\left(1+\frac{1}{\mathrm{a}+2}\right)\left(1+\frac{1}{\mathrm{a}+3}\right)\left(1+\frac{1}{\mathrm{a}+4}\right)$ _____.

If $a\times b=a+b+\frac{a}{b}$ then the value of $12\times 4$ is :

The value of $\frac{{\left(0.013\right)}^3+{\left(0.007\right)}^3}{{\left(0.013\right)}^2-0.013\times 0.007+{\left(0.007\right)}^2}$ is _____.

If, $x= 3+2\sqrt{2}$ , what is the value of $x^2+\frac{1}{x^2}$?

If $$\left( { x }^{ 2 }+\cfrac { 1 }{ { 5 } } x+{ a }^{ 2 } \right) $$ is a perfect square, then a is -

If $\mathrm{a}(\mathrm{a} + 2) = \mathrm{a} + \mathrm{b} + \mathrm{c}, \mathrm{b}(\mathrm{b} + 2) = \mathrm{a} + \mathrm{b} + \mathrm{c}$ and $\mathrm{c}(\mathrm{c}+2)=\mathrm{a}+\mathrm{b}+\mathrm{c}$, then what is the value of $\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}$?

If $x=2+\sqrt{3,} y=2-\sqrt{3},$ then the value $\frac{x^2+y^2}{x^3+y^3}$ is _____.

If $x = 3 +2\surd 2$ ,then the value of ${ x }^{ 4 }+\cfrac { 1 }{ { x }^{ 4 } }$ is

If $2x-\frac{1}{2x}=6$, then what is the value of $x^2+\frac{1}{16x^2}$?

$a \times b = b \times a$is

If ${\left(x+\frac{1}{x}\right)}^2=3$. then the value of $x^{206}+x^{200}+x^{90}+x^{84}+x^{12}+x^6+1$ is _____.

If ${\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2–\mathrm{ab}–\mathrm{bc}–\mathrm{ca}=0$, then _____.

If $\frac{{\left(x-y\right)}^2}{x^2-y^2}=9$ then, the value of $\frac{{\displaystyle \frac{x}{y}}+1}{{\displaystyle \frac{x}{y}}-1}$ is __________.