Find the values of the following expressions:

Solution:

(i) Let us consider the following expression:

$16 x^{2}+24 x+9$

now,

$16 x^{2}+24 x+9=(4 x+3)^{2} \quad\left(\right.$ Using identity $\left.(a+b)^{2}=a^{2}+2 a b+b^{2}\right)$

$\Rightarrow 16 x^{2}+24 x+9=\left(4 \times \frac{7}{4}+3\right)^{2} \quad\left(\right.$ Substituting $\left.x=\frac{7}{4}\right)$

$\Rightarrow 16 x^{2}+24 x+9=(7+3)^{2}$

$\Rightarrow 16 x^{2}+24 x+9=10^{2}$

$\Rightarrow 16 x^{2}+24 x+9=100$

(ii) Let us consider the following expression:

$64 x^{2}+81 y^{2}+144 x y$

Now,

$64 x^{2}+81 y^{2}+144 x y=(8 x+9 y)^{2} \quad$ (Using identity $(a+b)^{2}=a^{2}+2 a b+b^{2}$ )

$\Rightarrow 64 x^{2}+81 y^{2}+144 x y=\left[8(11)+9\left(\frac{4}{3}\right)\right]^{2} \quad$ (Substituting $x=11$ and $y=\frac{4}{3}$ )

$\Rightarrow 64 x^{2}+81 y^{2}+144 x y=[88+12]^{2}$

$\Rightarrow 64 x^{2}+81 y^{2}+144 x y=100^{2}$

$\Rightarrow 64 x^{2}+81 y^{2}+144 x y=10000$

(iii) Let us consider the following expression:

$81 x^{2}+16 y^{2}-72 x y$

Now,

$81 x^{2}+16 y^{2}-72 x y=(9 x-4 y)^{2} \quad\left(\right.$ Using identity $\left.(a+b)^{2}=a^{2}-2 a b+b^{2}\right)$

$\Rightarrow 81 x^{2}+16 y^{2}-72 x y=\left[9\left(\frac{2}{3}\right)-4\left(\frac{3}{4}\right)\right]^{2} \quad\left(\right.$ Substituting $x=\frac{2}{3}$ and $\left.y=\frac{3}{4}\right)$

$\Rightarrow 81 x^{2}+16 y^{2}-72 x y=[6-3]^{2}$

$\Rightarrow 81 x^{2}+16 y^{2}-72 x y=3^{2}$

$\Rightarrow 81 x^{2}+16 y^{2}-72 x y=9$