Ex.13.2 Q4 Exponents and Powers Solutions - NCERT Maths Class 7

Go back to  'Ex.13.2'

Question

Express each of the following as a product of prime factors only in exponential form:

(i) \(108 × 192 \)

(ii) \( 270\)

(iii) \(729 × 64\)

(iv) \(768\)

Text Solution

What is Known?

(i) \(108 × 192 \)

(ii) \( 270\)

(iii) \(729 × 64\)

(iv) \(768\)

What is unknown?

Product of prime factors only in exponential form.

Reasoning:

To solve this question, you must remember the laws of exponents. Here are some important laws of exponents

1.  \({{{a}}^{{m}}}{{ \times }}\,{{{a}}^{{n}}}= {{{a}}^{{{m + n}}}}\)

2.  \({{{a}}^{{m}}}{{ \div }}\,\,{{{a}}^{{n}}} = {{{a}}^{{{m - n}}}}\)

3.  \({{\left( {{{{a}}^{{m}}}} \right)}^{{n}}} = {{{a}}^{{{mn}}}}\)

4.  \({{{a}}^{{0}}}= 1\)

Steps:

\(\begin{align} \rm{(i)}\quad108{\rm{ }} &\times {\rm{ }}192\\&= {\rm{ }}2{\rm{ }} \times {\rm{ }}2 \times 2 \times 2 \times 2 \times 2 \times \,2 \times 2{\rm{ }} \times {\rm{ }}3{\rm{ }} \times {\rm{ }}3{\rm{ }} \times {\rm{ }}3{\rm{ }} \times 3\\&\qquad {a^m}\times {\rm{ }}{a^n} = {\rm{ }}{a^{m + n}}\\&= {\rm{ }}{2^8} \times {\rm{ }}{3^4}\end{align}\)

\(\begin{align}\left( {{\rm{ii}}} \right)270&\\&= {{ }}2{{ }} \times {{ }}3 \times {{ }}3{{ }} \times {{ }}3 \times 5 \qquad {{{a}}^{{m}}}{{ \times }}{{{a}}^{{n}}}\,{{ = }}\,{{{a}}^{{{m + n}}}}\\&= {{ }}2{{ }} \times {{ }}{3^3} \times {{ }}5 \\ &= 10 \times 3^3\end{align}\)

\(\begin{align}\left( {{\rm{iii}}} \right)\,\,729{\rm{ }} \times {\rm{ }}64&\\&= {\rm{ }}3 \times {\rm{ }}3 \times 3 \times 3 \times 3 \times 3 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\\&\qquad{a^m} \times {\rm{ }}{a^n} = {\rm{ }}{a^{m + n}}\\&= {\rm{ }}{3^6} \times {\rm{ }}{2^6}\\&= {\left( {3{\rm{ }} \times {\rm{ }}2} \right)^6}\end{align}\)

\(\begin{align}\left( {{\rm{iv}}} \right)768&\\&= {{ }}2{{ }} \times {{ }}2{{ }} \times {{ }}2{{ }} \times {{ }}2{{ }} \times {{ }}2{{ }} \times {{ }}2{{ }} \times {{ }}2{{ }} \times {{ }}2{{ }} \times {{ }}3\\&\qquad{{{a}}^{{m}}}{{ \times }}{{{a}}^{{n}}}{{ = }}\,{{{a}}^{{{m + n}}}}\\&= {{ }}{2^8} \times {{ }}3\end{align}\)

  
Learn from the best math teachers and top your exams

  • Live one on one classroom and doubt clearing
  • Practice worksheets in and after class for conceptual clarity
  • Personalized curriculum to keep up with school