Question

Express each of the following product as a monomials and verify the result in each case for x = 1:
(4x²) × (−3x) × \[\left( \frac{4}{5} x^3 \right)\]

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 law of indices, i.e.,​  \[a^m \times a^n = a^{m + n}\].

We  have:

\[\left( 4 x^2 \right) \times \left( - 3x \right) \times \left( \frac{4}{5} x^3 \right)\]

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

\[ = \left\{ 4 \times \left( - 3 \right) \times \frac{4}{5} \right\} \times \left( x^{2 + 1 + 3} \right)\]

\[ = - \frac{48}{5} x^6\]

\[\therefore\] \[\left( 4 x^2 \right) \times \left( - 3x \right) \times \left( \frac{4}{5} x^3 \right) = - \frac{48}{5} x^6\]

Substituting x = 1 in LHS, we get:

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

\[ = \left( 4 \times 1^2 \right) \times \left( - 3 \times 1 \right) \times \left( \frac{4}{5} \times 1^3 \right)\]

\[ = 4 \times \left( - 3 \right) \times \frac{4}{5}\]

\[ = - \frac{48}{5}\]

Putting x = 1 in RHS, we get:

\[\text { RHS } = - \frac{48}{5} x^6 \]

\[ = - \frac{48}{5} \times 1^6 \]

\[ = - \frac{48}{5}\]

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

Thus, the answer is \[- \frac{48}{5} x^6\].