Question

Express each of the following product as a monomials and verify the result in each case for x = 1:
(5x⁴) × (x²)³ × (2x)²

Answer 1

We have to find the product of the expression in order to express it as a monomial. 
To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices, i.e.,  \[a^m \times a^n = a^{m + n} \text { and } \left( a^m \right)^n = a^{mn}\]

We have:

\[\left( 5 x^4 \right) \times \left( x^2 \right)^3 \times \left( 2x \right)^2 \]

\[ = \left( 5 x^4 \right) \times \left( x^6 \right) \times \left( 2^2 \times x^2 \right)\]

\[ = \left( 5 \times 2^2 \right) \times \left( x^4 \times x^6 \times x^2 \right)\]

\[ = \left( 5 \times 2^2 \right) \times \left( x^{4 + 6 + 2} \right)\]

\[ = 20 x^{12} \]

\[\therefore\] \[\left( 5 x^4 \right) \times \left( x^2 \right)^3 \times \left( 2x \right)^2 = 20 x^{12} \]

Substituting x = 1 in LHS, we get:

\[\text { LHS } = \left( 5 x^4 \right) \times \left( x^2 \right)^3 \times \left( 2x \right)^2 \]

\[ = \left( 5 \times 1^4 \right) \times \left( 1^2 \right)^3 \times \left( 2 \times 1 \right)^2 \]

\[ = \left( 5 \times 1 \right) \times \left( 1^6 \right) \times \left( 2 \right)^2 \]

\[ = 5 \times 1 \times 4\]

\[ = 20\]

Put x =1 in RHS, we get:

\[RHS = 20 x^{12} \]

\[ = 20 \times \left( 1 \right)^{12} \]

\[ = 20 \times 1\]

\[ = 20\]

\[\because\] LHS = RHS for x = 1; therefore, the result is correct.

Thus, the answer is \[20 x^{12}\].