Express each of the following products as a monomial and verify the result for $x = 1,y = 2$:
$ \left(\frac{1}{8} x^{2} y^{4}\right) \times\left(\frac{1}{4} x^{4} y^{2}\right) \times(x y) \times 5 $

Given:

\( \left(\frac{1}{8} x^{2} y^{4}\right) \times\left(\frac{1}{4} x^{4} y^{2}\right) \times(x y) \times 5 \)

To do:

We have to express the given product as a monomial and verify the result for $x = 1,y = 2$.

Solution:

$(\frac{1}{8} x^{2} y^{4}) \times(\frac{1}{4} x^{4} y^{2}) \times(x y) \times 5=\frac{1}{8} \times \frac{1}{4} \times 5 \times x^{2} \times x^{4} \times x \times y^{4} \times y^{2} \times y$

$=\frac{5}{32} x^{2+4+1} \times y^{4+2+1}$

$=\frac{5}{32} x^{7} \times y^{7}$

$=\frac{5}{32} x^{7} y^{7}$

LHS $=(\frac{1}{8} x^{2} y^{4}) \times(\frac{1}{4} x^{4} y^{2}) \times(x y) \times 5$

$=\frac{1}{8} \times(1)^{2} \times(2)^{4} \times \frac{1}{4}(1)^{4}(2)^{2} \times 1 \times 2 \times 5$

$=\frac{1}{8} \times 1 \times 16 \times \frac{1}{4} \times 1 \times 4 \times 1 \times 2 \times 5$

$=\frac{640}{32}$

$=20$

RHS $=\frac{5}{32} x^{7} \times y^{7}$

$=\frac{5}{32}(1)^{7}(2)^{7}$

$=\frac{5}{32} \times 1 \times 128$

$=5 \times 4$

$=20$

Therefore,

LHS $=$ RHS