Find the best tutors and institutes for Class 10 Tuition
Q3(iv):
Simplify :
(iv) $7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}$
Solution :
Given Expression & Theoretical Foundation
We are tasked with simplifying the following mathematical expression involving rational exponents:
$7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}$
Upon analyzing the expression, we identify two distinct bases ($a = 7$ and $b = 8$) raised to an identical rational power ($m = \frac{1}{2}$).
Step 1: Invoking the Relevant Law of Exponents
To simplify the product of two different bases raised to the same exponent, we apply the Multiplicative Law of Exponents for Identical Powers. [Per the fundamental axioms of real number exponents], this law states:
$a^m \cdot b^m = (a \cdot b)^m$
This theorem holds true for any positive real numbers $a$ and $b$, and any rational number $m$.
Step 2: Algebraic Substitution and Simplification
By substituting our specific values into the established theorem ($a = 7$, $b = 8$, and $m = \frac{1}{2}$), we consolidate the bases inside a single parenthesis:
$7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}} = (7 \cdot 8)^{\frac{1}{2}}$
Next, we perform the arithmetic multiplication within the parentheses:
$7 \cdot 8 = 56$
Substituting this product back into the expression yields:
$56^{\frac{1}{2}}$
Step 3: Radical Conversion and Deep Simplification (Analytical Depth)
While $56^{\frac{1}{2}}$ is the simplified exponential form, it is mathematically rigorous to understand its equivalent radical form. [By the definition of rational exponents], an exponent of $\frac{1}{2}$ denotes the principal square root of the base:
$x^{\frac{1}{2}} = \sqrt{x}$
Therefore, our expression can be written as:
$\sqrt{56}$
To ensure the radical is in its simplest form, we extract the largest perfect square factor from the radicand (56). We perform prime factorization:
- $56 = 4 \cdot 14$
- $56 = 2^2 \cdot 14$
Applying the product property of radicals ($\sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y}$):
$\sqrt{56} = \sqrt{4 \cdot 14} = \sqrt{4} \cdot \sqrt{14} = 2\sqrt{14}$
Final Solution: The simplified expression is $56^{\frac{1}{2}}$ (which can also be rigorously expressed in radical form as $2\sqrt{14}$).
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(ii): Simplify : (ii) $(\frac{1}{3^3})^7$
- Q3(iii): Simplify : (iii) $\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Number Systems
I teach Mathematics to students from Class 7 to Class 10, with 3 years of teaching experience. My focus is on helping students build strong conceptual understanding and develop effective problem-solving skills to excel academically. I have completed my 2nd PUC (Pre-University Course), a Bachelor of Engineering (B.E.) from Visveswaraya Technological University in 2020, and a Bachelor of Education (B.Ed.) from Indira Gandhi National Open University in 2022. I have also completed a program with Cambridge Learning Resources, which has enhanced my teaching techniques and ability to engage students effectively. Over the years, I have guided students to improve their Mathematics understanding and performance, helping them gain confidence in handling both basic and complex problems. My teaching approach emphasizes clarity, practice, and logical reasoning to ensure steady improvement. I have also earned an RPA (Robotic Process Automation) certification in robotics, which reflects my technical skills and commitment to continuous learning. I often incorporate logical thinking and problem-solving strategies inspired by this training into my lessons. I prefer to conduct online classes, offering flexibility in timing and easy access for students. I teach both individual and small group sessions, allowing personalized attention and interactive learning experiences for every student. My classes are interactive, using motivational videos, personal digital pads, and other digital tools to make learning engaging and effective. These resources help students visualize concepts better and retain knowledge more efficiently. I provide custom-prepared notes to support student learning and revision. My ultimate goal is to make Mathematics understandable, enjoyable, and confidence-building, enabling every student to perform at their best and develop a strong foundation for future studies.
This guy is too good at his teaching. He explains the concepts very clear, gets to know about the students, gives his own ideas, own methods, and also gives good advices. He is excellent to consider. I just like him!, he is a great tutor and price is reasonably good.
10+ years of experience teaching Mathematics for Grades 10 to 12th • Consistent record of 100% academic results • Very friendly and supportive learning environment • Encourages students to ask questions freely and confidently • Strong focus on conceptual clarity and problem-solving skills • Helps students build confidence and excel in Mathematics • Fee Structure for One-to-One Classes: ₹700 per hour • Group Classes: ₹4,000 per month.
Sir clears all my doubts he makes difficult concepts easy to understand he takes Previous year questions which makes it easier to prepare
I have Taught 3 years in CBSE board and 5 years In ICSE board in class 10 with 100% results.Got an opportunity to go for CBSE board class 10 Maths copy correction.Was Board examiner head for ICSE .
Pragya ma'am is amazing! She helped me with maths a lot! Great teacher!
I have a total of 10 years of teaching experience. I am Teaching Online for Overseas and Local Students. My Students are from INDIA, UK, USA, INDONESIA & THAILAND. My passion is Teaching. I am Focusing on CBSE, IGCSE, IB, ICSE,&NIOS,.I have conducted National level conference and workshop, Hands-on-training, Seminar, etc Teaching with Human touch produces a great result. My students of class 10th had proven with good results.
I started my teaching career in 2005 in that year itself I got a chance to take class in 10 for one month for leave vacancy. After that I got the chance in 2011 in another school, then it continued. Taking class in 10 it was a challenge because responsibilities are more. Both parents and students are in stress, at the same time the school management also. If the results are not satisfactory it will affect to the reputation of the school. So the teachers are alert always .All the students have to face the board. So the preparation of each and every child is important. For that conducting special classes ,morning classes. Conducting evaluations in proper interval. After every evaluation a session with parents in school is very important. Because they are more stressed than students. The year tenth is important because it is the turning point of each student . The effort of all the pupils will be fruitful when we get the good result. Teachers especially those who are taking in class 10 they are not bothered about the salary which the management as paying to them, of course it is a pain to their personal life. Once if we are standing in front of the students our main attention was that whether the concept will be reaching to the pupil in the right meaning and sense. In that time we are not bothered about the payment. If the student got the concept and scored good in exam that was the great reward for teacher. After the schooling that student and parents never forget that teacher in their life every important movements they expect that teachers presence there.
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
Download free CBSE - Class 9 Mathematics Number Systems EXERCISE 1.5 worksheets
Download Now