Find the best tutors and institutes for Class 10 Tuition
Q3(ii):
Simplify :
(ii) $(\frac{1}{3^3})^7$
Solution :
Initial Setup & Given Expression
We are tasked with simplifying the following exponential expression:
$ E = \left(\frac{1}{3^3}\right)^7 $
To simplify this expression rigorously, we will apply the fundamental Laws of Exponents. There are two primary analytical pathways to arrive at the solution. Both methods are detailed below to ensure comprehensive understanding.
Method 1: Applying the Negative Exponent Rule First
Step 1: Transform the Fractional Base
[Per the Reciprocal Law of Exponents], any non-zero base raised to a positive power in the denominator can be expressed as a negative exponent in the numerator. The governing axiom is:
$ \frac{1}{a^n} = a^{-n} $
Applying this theorem to the inner fraction of our expression:
$ \frac{1}{3^3} = 3^{-3} $
Step 2: Substitute and Apply the Power of a Power Rule
Substitute the transformed base back into the original expression:
$ E = \left(3^{-3}\right)^7 $
[Per the Power of a Power Property], when an exponential term is raised to another exponent, the exponents are multiplied. The governing axiom is:
$ (a^m)^n = a^{m \cdot n} $
Applying this property to our expression yields:
$ E = 3^{-3 \cdot 7} $
Step 3: Arithmetic Simplification
Perform the multiplication in the exponent:
$ -3 \cdot 7 = -21 $
$ E = 3^{-21} $
Method 2: Applying the Power of a Quotient Rule First
Step 1: Distribute the Exponent
[Per the Power of a Quotient Rule], an exponent applied to a fraction is distributed to both the numerator and the denominator. The governing axiom is:
$ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $
Applying this to our expression:
$ E = \frac{1^7}{(3^3)^7} $
Step 2: Simplify Numerator and Denominator
Since $1$ raised to any finite power is $1$ ($1^7 = 1$), the numerator simplifies immediately. For the denominator, we again apply the Power of a Power Property $(a^m)^n = a^{m \cdot n}$:
$ E = \frac{1}{3^{3 \cdot 7}} $
$ E = \frac{1}{3^{21}} $
Step 3: Convert to a Negative Exponent
Using the Reciprocal Law of Exponents $\left(\frac{1}{a^n} = a^{-n}\right)$ to express the final answer without a fraction:
$ E = 3^{-21} $
Visual Representation of the Exponent Laws
The following diagram illustrates the sequential application of the exponent rules used in Method 1.
Final Solution: The simplified expression is $3^{-21}$ (which can also be written as $\frac{1}{3^{21}}$).
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.5
- Q1(i): Find : (i) $64^{\frac{1}{2}}$
- Q1(ii): Find : (ii) $32^{\frac{1}{5}}$
- Q1(iii): Find : (iii) $125^{\frac{1}{3}}$
- Q2(i): Find : (i) $9^{\frac{3}{2}}$
- Q2(ii): Find : (ii) $32^{\frac{2}{5}}$
- Q2(iii): Find : (iii) $16^{\frac{3}{4}}$
- Q2(iv): Find : (iv) $125^{-\frac{1}{3}}$
- Q3(i): Simplify : (i) $2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}$
- Q3(iii): Simplify : (iii) $\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}$
- Q3(iv): Simplify : (iv) $7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Number Systems
Top Tutor
Sultanpur Viran, Sultanpur Viran
9 yrs of
Exp
9 students
1105 hrs of classes completed
1000
per hour
Online Classes
Student's Home
Tutor's home
I have a teaching experience of 9+ years. Currently teaching in St. Mary's Convent School. My way of teaching is simple and friendly as per the requirement. I lay emphasis on the basics because that only helps to make the concepts more easier to understand.
Excellent teacher. He should be given lot of work. Very kind person. He took my daughter from lowest level to 80% in a very short period.
Super Tutor
Sector 5, Dwarka, Delhi
10 yrs of
Exp
13 students
1261 hrs of classes completed
600
per hour
Online Classes
I teach Math and Science to 8th-12th standard students and have been teaching for more than 10 years. Batch of 2 Students: I conduct these small batches with a pair of students of the same class and same education board (I cover all popular boards CBSE, ICSE, IGCSE etc.). This works for those students/parents who are looking for individual attention for their kids but still want it to be a little affordable. *3 classes per week (So 12 a month) *Maths and Science (Also English Grammar and Literature) *Fee Rs. 4500/- per month/12 classes. Batch of 6 Students: I conduct these small batches with 5-6 students of the same class and same education board (I cover all popular boards CBSE, ICSE, IGCSE etc.). *4 classes per week (So 16 a month) *Maths and Science *Fee Rs. 4000/- per month/16 classes. I will start the classes immediately once the first kid enrolls. -------------------------------------------------------------------------------Individual/Private/One-to-One classes: I conduct these one to one classes for those students willing to have a one-to-one setup (I cover all popular boards CBSE, ICSE, IGCSE etc.). *3 classes per week (So 12 a month) *Maths and Science (Also English Grammar and Literature) *Fee Rs. 7000/- per month/12 classes. I will start the classes immediately once the first kid enrolls. -------------------------------------------------------------------------------I took classes and batches in different institutions.
The experience with virendra sir was amazing. He did away with all my doubts and difficulties at all times , he is more than a teacher and a mentor to me. I'll always remember sir's debate's , scolding and care for me. Ty sir for making me a greater individual.
Top Tutor
Nerul, Mumbai
6 yrs of
Exp
1 student
347 hrs of classes completed
400
per hour
Online Classes
Student's Home
Tutor's home
I have specifically guided Class 10 students for board exams, with a strong focus on NCERT syllabus, and topics like Real Numbers, Polynomials, Quadratic Equations, and Statistics. I provide regular tests, personalized feedback, and exam-oriented strategies to improve accuracy and speed. Many of my students have shown remarkable improvement and scored above 90% in their board exams.
Sir is very nice in explaining the concepts . He gives many question for me to practice and explains them in a simple and straightforward manner.
Top Tutor
Kamal Chowk, Nagpur
20 yrs of
Exp
Tutor's home
I have 20 years of experience in the field of teaching. I conduct classes for Spoken English, BBA, MBA, Hindi Language, Career Counselling and Classes I to X. I tutor (offline/online) for syllabi-based subjects Classes I to X for CBSE, ICSE, IGCSE, IB Board. For classes X and above, I focus on tutoring subjects like Hindi, English, Psychology. Accountancy, Business Studies, Mathematics, Physics, Chemistry for CBSE, ICSE, IGCSE and IB Board. I also conduct career counselling and psychological counselling. I also assist students in pursuing MBA/PGDBM courses in their dissertation work. People engaged in doctoral theses in the domains of management and psychology engage my services for their research work. Since there is a lot of demand for enhancing one's conversational English, this is what I am focusing on at the moment. Being bilingual, my medium of communication is English as well as Hindi. I train learners in the usage of WordPress, Joomla - to develop their websites; right from domain registration to site maintenance. Besides, I assist teachers in pursuing Teachers' Training Courses.
Top Tutor
Padmanabhanagar, Bangalore
3 yrs of
Exp
450
per hour
Online Classes
Student's Home
Tutor's home
I am an experienced tutor with 4 years of experience in this field. I completed my BE in the year 2014. I have tutored students for Spoken English, BTech Tuition, C Language, C++ Language, CAD, CE and COMEDK Coaching. I can conduct classes at Student's Home, My place and Online via a different medium to students from International, CBSE, State, ICSE/ISC Board. I follow different techniques while teaching so, that my students are engaged positively. To make learning fun and easier, I use creative and innovative ideas.
Find more Tutor for Number Systems in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
