Applications of Identities

Expansion of ${\left(5{\mathrm{a}}^2+6{\mathrm{b}}^2\right)}^2$ is

Consider, the special case of identity $(x+a)(x+b)=x^2+(a+b)x+ab$ with$a=b$, what do you get? Is it related to identity, ${\left(a+b\right)}^2=a^2+2ab+b^2$?

Find: $\frac{55\times 55-40\times 40}{55+40}$

If $\left(x-y\right)=5$ and $xy=6$, then the value of $x^2+y^2$ is

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

If $\left(x+y\right)=5$ and $xy=6$, find the value of $x^2-y^2$.

If $\left(x+y\right)=9$ and $xy=14$, find the value of $x^2-y^2$.

If $x-\frac{1}{x}=4$, find the value of $x^4+\frac{1}{x^4}$

If $x-\frac{1}{x}=4$, find the value of $x^2+\frac{1}{x^2}$

The product of $\left(x+2y\right)\left(x-2y\right)\left(x^2+4y^2\right)$ is $x^4-p{y}^4$, find the value of $p$.

The product of $\left(x+y\right)\left(x-y\right)\left(x^2+y^2\right)$ is $x^p-y^q$, find the vlaue of $p+q$

Find the value of the following expression.

$x^2+6xy+81y^2$, when $x=-1$, $y=2$

Find the value of the following expression.

$16x^2-\frac{16}{3}xy+\frac{4}{9}{y}^2$, when $x=\frac{1}{3}$ and $y=\frac{1}{2}$

Find the value of the following expression.

$81x^2-144x+64$, when $x=-2$

Find the value of the following expression.

$16a^2+40a+25$, when $a=2$

Simplify:

$\frac{105\times 105-5\times 5}{2}$

Simplify:

$\frac{297\times 297-203\times 203}{94}$

Simplify:

$\frac{16.2\times 16.2-2\left(16.2\right)\left(4.2\right)+4.2\times 4.2}{3.5\times 3.5-0.5\times 0.5}$

Using the identities find the value of the following.

$10.3\times 8.7$

 If $x^2+y^2=13$ and $xy=6$, find the value of the following.

$x-y$