Question

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

Answer 1

To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices, i.e.,​ $a^m\times a^n=a^{m+n}$.

We have:

$$\begin{gathered} \left(\frac{1}{8}{x}^2{y}^4\right)\times \left(\frac{1}{4}{x}^4{y}^2\right)\times \left(xy\right)\times 5 \\ =\left(\frac{1}{8}\times \frac{1}{4}\times 5\right)\times \left(x^2\times x^4\times x\right)\times \left(y^4\times y^2\times y\right) \\ =\left(\frac{1}{8}\times \frac{1}{4}\times 5\right)\times \left(x^{2+4+1}\right)\times \left(y^{4+2+1}\right) \\ =\frac{5}{32}{x}^7{y}^7 \end{gathered}$$

To verify the result, we substitute x = 1 and y = 2 in LHS; we get:

$$\begin{gathered} \text{LHS}=\left(\frac{1}{8}{x}^2{y}^4\right)\times \left(\frac{1}{4}{x}^4{y}^2\right)\times \left(xy\right)\times 5 \\ =\left\{\frac{1}{8}\times {\left(1\right)}^2\times {\left(2\right)}^4\right\}\times \left\{\frac{1}{4}\times {\left(1\right)}^4\times {\left(2\right)}^2\right\}\times \left(1\times 2\right)\times 5 \\ =\left(\frac{1}{8}\times 1\times 16\right)\times \left(\frac{1}{4}\times 1\times 4\right)\times \left(1\times 2\right)\times 5 \\ =2\times 1\times 2\times 5 \\ =20 \end{gathered}$$

Substituting x = 1 and y = 2 in RHS, we get:​

$$\begin{gathered} \text{RHS}=\frac{5}{32}{x}^7{y}^7 \\ =\frac{5}{32}{\left(1\right)}^7{\left(2\right)}^7 \\ =\frac{5}{\cancel{32}}\times 1\times {\cancel{128}}^4 \\ =20 \end{gathered}$$

Because LHS is equal to RHS, the result is correct.

Thus, the answer is $\frac{5}{32}{x}^7{y}^7$.