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:

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

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

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

\[ = \frac{5}{32} x^7 y^7\]

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

\[\text { LHS } = \left( \frac{1}{8} x^2 y^4 \right) \times \left( \frac{1}{4} x^4 y^2 \right) \times \left( xy \right) \times 5\]

\[ = \left\{ \frac{1}{8} \times \left( 1 \right)^2 \times \left( 2 \right)^4 \right\} \times \left\{ \frac{1}{4} \times \left( 1 \right)^4 \times \left( 2 \right)^2 \right\} \times \left( 1 \times 2 \right) \times 5\]

\[ = \left( \frac{1}{8} \times 1 \times 16 \right) \times \left( \frac{1}{4} \times 1 \times 4 \right) \times \left( 1 \times 2 \right) \times 5\]

\[ = 2 \times 1 \times 2 \times 5\]

\[ = 20\]

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

\[\text { RHS } = \frac{5}{32} x^7 y^7 \]

\[ = \frac{5}{32} \left( 1 \right)^7 \left( 2 \right)^7 \]

\[ = \frac{5}{32} \times 1 \times {128}^4 \]

\[ = 20\]

Because LHS is equal to RHS, the result is correct.
Thus, the answer is \[\frac{5}{32} x^7 y^7\].