Question

Find the product of \[x\],\[{x^2}\],\[{x^3}\],\[{x^4}\]

Answer 1

Hint: First we have to know what surds and indices. Mention their laws. Then using laws of surds to find the product by taking the product of the product of first two and last two values.

Complete step-by-step solution:
Given \[x\],\[{x^2}\],\[{x^3}\],\[{x^4}\]----(1)
Then the product of first two terms \[x\]and \[{x^2}\] using the formula \[{a^m} \times {a^n} = {a^{m + n}}\] is
\[x \times {x^2} = {x^{1 + 2}} = {x^3}\]----(2)
Then the product of last two terms \[{x^3}\]and \[{x^4}\] using the formula \[{a^m} \times {a^n} = {a^{m + n}}\] is
\[{x^3} \times {x^4} = {x^{3 + 4}} = {x^7}\]----(3)
Then the product of \[{x^3}\]and \[{x^7}\](form the equations (2) and (3) ) is equal to the product of \[x\],\[{x^2}\],\[{x^3}\],\[{x^4}\], using the formula \[{a^m} \times {a^n} = {a^{m + n}}\] is
\[{x^3} \times {x^7} = {x^{3 + 7}} = {x^{10}}\]----(3)

Hence, the product of \[x\],\[{x^2}\],\[{x^3}\],\[{x^4}\] is \[{x^{10}}\].


Note: Laws of indices: Suppose \[a\]and \[b\]are any two non-zero real numbers. Let \[n\] and \[m\] be any two non-zero integers then the following laws holds:
- \[{a^m} \times {a^n} = {a^{m + n}}\]
- \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\]
- \[{\left( {{a^m}} \right)^n} = {a^{mn}}\]
- \[{\left( {ab} \right)^n} = {a^n}{b^n}\]
- \[{\left( {\dfrac{a}{b}} \right)^n} = \dfrac{{{a^n}}}{{{b^n}}}\]
- \[{a^0} = 1\].