Find the best tutors and institutes for Class 10 Tuition
Q1(i):
Find :
(i) $64^{\frac{1}{2}}$
Solution :
Initial Setup & Given Expression
We are tasked with evaluating the following exponential expression:
$64^{\frac{1}{2}}$
Step 1: Theoretical Foundation of Fractional Exponents
[Per the Definition of Rational Exponents], an expression of the form $a^{\frac{m}{n}}$ is mathematically equivalent to the $n$-th root of $a$ raised to the $m$-th power, expressed as $\sqrt[n]{a^m}$.
In our specific case, the exponent is $\frac{1}{2}$. This indicates the principal square root of the base. Therefore:
$64^{\frac{1}{2}} = \sqrt[2]{64^1} = \sqrt{64}$
Step 2: Method 1 - Prime Factorization (Rigorous Approach)
To evaluate this expression algebraically without relying on memorized perfect squares, we first determine the prime factorization of the base, $64$.
- $64 \div 2 = 32$
- $32 \div 2 = 16$
- $16 \div 2 = 8$
- $8 \div 2 = 4$
- $4 \div 2 = 2$
- $2 \div 2 = 1$
Thus, the prime factorization is $64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6$.
Substituting this back into our original expression yields:
$(2^6)^{\frac{1}{2}}$
Step 3: Applying the Power of a Power Property
[Per the Laws of Exponents], specifically the Power of a Power property which states that $(a^m)^n = a^{m \cdot n}$, we multiply the inner exponent by the outer fractional exponent:
$(2^6)^{\frac{1}{2}} = 2^{6 \times \frac{1}{2}}$
$2^{6 \times \frac{1}{2}} = 2^3$
Evaluating $2^3$ gives:
$2 \times 2 \times 2 = 8$
Step 4: Method 2 - Perfect Square Identification (Direct Approach)
Alternatively, recognizing that $64$ is a perfect square provides a more direct algebraic route. We know that $8 \times 8 = 64$, meaning $64 = 8^2$.
Substituting $8^2$ into the expression:
$(8^2)^{\frac{1}{2}}$
Applying the same Power of a Power property $(a^m)^n = a^{m \cdot n}$:
$8^{2 \times \frac{1}{2}} = 8^1 = 8$
Geometric Representation of the Square Root
Geometrically, finding the square root of $64$ (i.e., $64^{\frac{1}{2}}$) is equivalent to finding the side length of a square whose total area is $64$ square units. The diagram below illustrates an $8 \times 8$ grid, visually proving that a side length of $8$ yields an area of $64$.
Final Conclusion
Through both prime factorization and perfect square identification, governed by the fundamental laws of exponents, the expression simplifies perfectly to the integer $8$.
Final Solution: 8
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.5
- 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(ii): Simplify : (ii) $(\frac{1}{3^3})^7$
- 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
Ayodhya Nagar, Vijayawada
5 yrs of
Exp
1 student
713 hrs of classes completed
470
per hour
Online Classes
I am an experienced educator with 8 years of teaching and training expertise, specializing in all subjects for students from Class 1 to 8 across various academic boards. Additionally, I offer professional Microsoft Excel training (basic to advanced), designed to equip learners with practical skills for academic and professional use. I hold a B.Tech in Information Technology, which has strengthened my analytical and technical foundation, enabling me to integrate practical and modern methods into my teaching. Over the past 8 years, I have worked with students from diverse backgrounds and learning levels, ensuring that every learner gets individual attention and a personalized learning experience. I strongly believe in active communication with both students and parents, addressing concerns and adapting methods to suit each child’s unique pace and goals. One of my proudest achievements is the academic success of my students—for instance, 8 out of 10 students recently scored 95+ in Mathematics under my guidance. I was also recognized as a Star Tutor for 6 consecutive months at Filo EdTech, based on session quality, student feedback, and consistent results, which reflects my dedication to quality education. I conduct only online classes for flexible and convenient learning. My teaching approach is structured yet interactive, ensuring clarity and engagement through the following steps: Concept Explanation: Breaking down topics into simple, easy-to-understand concepts. Interactive Sessions: Encouraging students to ask questions and clarify doubts during class. Practice Work: Providing assignments and exercises for reinforcement. Progress Tracking: Offering regular feedback and performance assessments. For Microsoft Excel training, I follow a practical-based approach, using real-life examples and hands-on exercises, so learners can immediately apply their knowledge. I make use of PowerPoint presentations, digital whiteboards, and provide notes when required to ensure an effective and engaging learning experience.
Top Tutor
Sikandra, Agra
4 yrs of
Exp
1 student
350
per hour
Online Classes
Student's Home
I can teach every topic very easily and make it stronger for the students.
Top Tutor
Shyam Nagar Vivek Vihar, Jaipur
10 yrs of
Exp
3 students
174 hrs of classes completed
150
per hour
Online Classes
Tutor's home
I am a teacher of Mathematics. I am teaching in school since 2013. I have a degree in Education. I have done Masters in Mathematics.
Have excellent method of teaching. Solve all doubts in easy way. Good command on subject and every topic.
Top Tutor
Uttam Nagar, Delhi
9 yrs of
Exp
5 students
312 hrs of classes completed
300
per hour
Online Classes
Tutor's home
teach them with practical and theory base method
Super Tutor
Kharghar Sector 27, Mumbai
13 yrs of
Exp
30 students
1078 hrs of classes completed
650
per hour
Online Classes
Student's Home
**Teaching Methodology:** My approach to teaching Mathematics and Science is personalized and interactive. I focus on understanding each student's unique learning style and adapt my methods accordingly. My goal is to make Maths and Science relatable and engaging, using real-life examples and practical applications to enhance understanding.I try to develop self thinking on problem and pushing them to solve by self **Key Skills:** - Strong grasp of Mathematics concepts across various levels - Ability to simplify complex topics for better understanding - Excellent communication and motivational skills - Patient and supportive, ensuring a comfortable learning environment - Regular progress assessments and constructive feedback **Services Offered:** - One-on-one personalized tutoring sessions - Homework and assignment assistance - Exam preparation and revision classes - Online and in-person tutoring options - Flexible scheduling to accommodate student needs **Achievements:** - Helped students improve their grades by an average of [25%] - Successfully prepared students for competitive exams with [JEE,NEET,IMO] - Received positive feedback and high ratings from students and parents Interested in improving your Maths AND Science skills? Contact me today to schedule a session and take the first step towards mastering Mathematics and Science!
This teacher is incredibly friendly and teaches in a very calm and patient manner. He helped me clear all my doubts; after failing my board exams previously, I managed to score 47/80 in Mathematics under his guidance. He truly excels at building a student's confidence.
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
