Question

Find the Quotient of: \[\dfrac{{ - 18{h^4}{k^5}}}{{ - 6{h^2}{k^2}}}\]

Answer 1

Hint:
Quotient refers to the term obtained by dividing one term by another. The term which is being divided is known as a dividend, whereas the term by which it is being divided is known as the divisor. When divisor and dividend are represented by fraction, dividend lies above the fraction known as the numerator, whereas the divisor lies below the fraction known as the denominator. \[fraction = \dfrac{{dividend}}{{divisor}} = \dfrac{{numerator}}{{quatient}}\]
In the question, we can see numerator and denominator both have constant terms, which are the same and have some power to it, which are different. We will use the power of quotient property, which in general says that when the numerator and denominator have the same constant terms having some raised power on them, then the constant terms raised the power of the denominator are transferred to the raised power of numerator with the same constant terms where raised power are subtracted.
\[\dfrac{{{a^5}}}{{{a^3}}} = {a^{5 - 3}} = {a^2}\]
When the term in the denominator does not lie in the numerator, then the raised power is inverse\[\left( {\dfrac{1}{{{a^m}}} = {a^{ - m}}} \right)\]
Complete step by step solution:
Divide \[ - 18{h^4}{k^5}\] by \[ - 6{h^2}{k^2}\] where \[ - 18{h^4}{k^5}\] is the dividend will lie on the numerator and \[ - 6{h^2}{k^2}\]is the divisor, which lies on the denominator.
We can see the constant term \[h\], and \[k\] both lie in the numerator and the denominator has raised power to them.
Now move the constant terms of the denominator with raised power to the numerator and subtract the raised power from the raised power of constant terms,
\[\dfrac{{ - 18{h^4}{k^5}}}{{ - 6{h^2}{k^2}}} = \dfrac{{ - 18}}{{ - 6}}\left( {{h^4}{k^5} \times {h^{ - 2}}{k^{ - 2}}} \right) = 3\left( {{h^{4 - 2}}{k^{5 - 2}}} \right) = 3\left( {{h^2}{k^3}} \right)\]
hence quotient we get is
 \[\dfrac{{ - 18{h^4}{k^5}}}{{ - 6{h^2}{k^2}}} = 3\left( {{h^2}{k^3}} \right)\]

Note:
In general, in the case of the division, when the raised powers of the denominator are transferred to the numerator, then the raised powers are subtracted, and in the case of multiplication, the raised power is added.