Question

Carry out the following divisions
I.\[28{x^4} \div 56x\]
II.\[ - 36{y^3} \div 9{y^2}\]
III.\[66p{q^2}{r^3} \div 11q{r^2}\]
IV.\[34{x^3}{y^3}{z^3} \div 51x{y^2}{z^3}\]
V.\[12{a^8}{b^8} \div \left( { - 6{a^6}{b^4}} \right)\]

Answer 1

Hint: Use the formula \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\]for the division identical variables and the numbers.
The division is a process of dividing one number by another used for dividing a matrix, vector, or the other quantity under specific rules.
Each party involved in a division equation has a special name where the dividend is the number that is being divided, a divisor is a number by which dividend is divided, whereas the quotient is the result obtained in the division process.

Complete step-by-step answer:
I.\[28{x^4} \div 56x\], can be written as \[\dfrac{{28{x^4}}}{{56x}}\]
Now divide
\[
   = \dfrac{{28}}{{56}} \times \dfrac{{{x^4}}}{x} \\
   = \dfrac{1}{2} \times \dfrac{{{x^4}}}{{{x^1}}} \\
 \]
Now by using the exponent division formula, we get
\[ = \dfrac{1}{2} \times {x^{4 - 1}}\]
This is equals to
\[ = \dfrac{1}{2} \times {x^3}\]
Hence \[28{x^4} \div 56x = \dfrac{1}{2} \times {x^3}\]

II.\[ - 36{y^3} \div 9{y^2}\], can be written as \[\dfrac{{ - 36{y^3}}}{{9{y^2}}}\]
Now divide
\[
   = - \dfrac{{36}}{9} \times \dfrac{{{y^3}}}{{{y^2}}} \\
   = - 4 \times \dfrac{{{y^3}}}{{{y^2}}} \\
 \]
Now by using the exponent division formula, we get
\[ = - 4 \times {y^{3 - 2}}\]
This is equal to
\[ = - 4{y^1}\]
Hence\[ - 36{y^3} \div 9{y^2} = - 4y\]

III.\[66p{q^2}{r^3} \div 11q{r^2}\], can be written as \[ = \dfrac{{66p{q^2}{r^3}}}{{11q{r^2}}}\]
Now divide
\[
   = \dfrac{{66}}{{11}} \times p \times \dfrac{{{q^2}}}{q} \times \dfrac{{{r^3}}}{{{r^2}}} \\
   = 6 \times p \times \dfrac{{{q^2}}}{{{q^1}}} \times \dfrac{{{r^3}}}{{{r^2}}} \\
 \]
Now by using the exponent division formula, we get
\[ = 6 \times p \times {q^{2 - 1}} \times {r^{3 - 2}}\]
This is equal to
\[ = 6pqr\]
Hence\[66p{q^2}{r^3} \div 11q{r^2} = 6pqr\]

IV.\[34{x^3}{y^3}{z^3} \div 51x{y^2}{z^3}\], can be written as \[ = \dfrac{{34{x^3}{y^3}{z^3}}}{{51x{y^2}{z^3}}}\]
Now divide
\[
   = \dfrac{{34}}{{51}} \times \dfrac{{{x^3}}}{x} \times \dfrac{{{y^3}}}{{{y^2}}} \times \dfrac{{{z^3}}}{{{z^3}}} \\
   = \dfrac{2}{3} \times \dfrac{{{x^3}}}{{{x^1}}} \times \dfrac{{{y^3}}}{{{y^2}}} \times \dfrac{{{z^3}}}{{{z^3}}} \\
 \]
Now by using the exponent division formula, we get
\[ = \dfrac{2}{3} \times {x^{3 - 1}} \times {y^{3 - 2}} \times {z^{3 - 3}}\]
This is equal to
\[ = \dfrac{2}{3}{x^2}{y^1}{z^0}\]
Hence\[34{x^3}{y^3}{z^3} \div 51x{y^2}{z^3} = \dfrac{2}{3}{x^2}y\]

V.\[12{a^8}{b^8} \div \left( { - 6{a^6}{b^4}} \right)\], can be written as \[ = \dfrac{{12{a^8}{b^8}}}{{ - 6{a^6}{b^4}}}\]
Now divide
\[
   = - \dfrac{{12}}{6} \times \dfrac{{{a^8}}}{{{a^6}}} \times \dfrac{{{b^8}}}{{{b^4}}} \\
   = - 2 \times \dfrac{{{a^8}}}{{{a^6}}} \times \dfrac{{{b^8}}}{{{b^4}}} \\
 \]
Now by using the exponent division formula, we get
\[ = - 2 \times {a^{8 - 6}} \times {b^{8 - 4}}\]
This is equal to
\[ = - 2{a^2}{b^4}\]
Hence\[12{a^8}{b^8} \div \left( { - 6{a^6}{b^4}} \right) = - 2{a^2}{b^4}\]

Note: Students are advised to be careful while using the exponential division formula as, the numbers whose base is the same can only be applied through the exponential division formula. A divisor and a dividend can be represented in terms of factor where the numerator of the fraction is the dividend, and the denominator is the divisor is represented
\[\dfrac{x}{y} = \dfrac{{numerator}}{{deno\min ator}} = \dfrac{{dividend}}{{divisor}}\]
In this question, the division consist of the numbers and the variables, and division of variables is not possible; hence use formula\[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\], where ‘a’ is the variable and m and n are exponents to carry out the division.