Math Problem Statement
Solution
Let's break down the problem in the image.
You are given the following prime factorizations of A, B, and C:
- A =
- B =
- C =
The tasks are:
- Find the value of the square root of A.
- Find the lowest common multiple (LCM) of A, B, and C.
- Find the highest common factor (HCF) of A, B, and C.
Step-by-step solutions:
(i) The value of the square root of A:
To find the square root of A, you will take the square root of each prime factor with its exponent:
- A =
The square root of a product is the product of the square roots of each factor. So:
(ii) The lowest common multiple (LCM) of A, B, and C:
The LCM is found by taking the highest power of each prime factor that appears in any of the numbers.
- For , the highest power is (from B and C).
- For , the highest power is (from B).
- For , the highest power is (from C).
- For , the highest power is (from C).
So, the LCM is:
(iii) The highest common factor (HCF) of A, B, and C:
The HCF is found by taking the lowest power of each prime factor that appears in all of the numbers.
- For , the lowest power is (from A).
- For , the lowest power is (from A and B).
- The primes 5 and 7 do not appear in all three numbers, so they are not part of the HCF.
So, the HCF is:
Final Answers:
- The square root of A is .
- The LCM of A, B, and C is .
- The HCF of A, B, and C is .
Let me know if you need further clarification or details on any of the steps!
Related Questions:
- How do you calculate the square root of a number written in prime factorization form?
- What is the method to find the least common multiple (LCM) using prime factorization?
- How do you find the highest common factor (HCF) using prime factorization?
- Can you calculate the square root of any prime factorization?
- What would happen if the prime factors of A, B, and C were different?
Tip:
When finding the LCM or HCF using prime factorizations, always remember:
- For LCM, take the highest power of each prime factor.
- For HCF, take the lowest power of each common prime factor.
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Math Problem Analysis
Mathematical Concepts
Prime Factorization
Square Roots
LCM (Least Common Multiple)
HCF (Highest Common Factor)
Formulas
Square root of a number: √(a × b) = √a × √b
LCM: Take the highest powers of all primes in the factorization
HCF: Take the lowest powers of all common primes in the factorization
Theorems
Prime Factorization Theorem
LCM and HCF based on prime factors
Suitable Grade Level
Grades 7-9
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