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If 3 ^ a = 5 ^ b = (75) ^ c then the value of ab - c(2a b) reduces to?
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If 3 ^ a = 5 ^ b = (75) ^ c then the value of ab - c(2a b) reduces t...
Solution:
Given, 3^a = 5^b = (75)^c
We know that 75 = 3^1 × 5^2
∴ (75)^c = (3^1 × 5^2)^c = 3^c × 5^(2c)
So, we can write 3^a = 5^b = 3^c × 5^(2c)
Let us take two cases:
Case I: When a = b
If a = b, then 3^a = 5^b
∴ 3^a = 3^c × 5^(2c) and 5^b = 3^c × 5^(2c)
∴ 3^a = 5^b
⇒ 3^a/5^b = 1
⇒ 3^(a-b) = 1
⇒ a - b = 0
Now, ab - c(2a - b) = ab - 2ac + bc
⇒ b(a - 2c) + ac
⇒ b(a - 2c) + b(a - b)
⇒ b(a - b - 2c)
Since a = b, the above expression becomes:
⇒ b(a - a - 2c) = -2bc
Therefore, ab - c(2a - b) reduces to -2bc.
Case II: When a ≠ b
If a ≠ b, then 3^a ≠ 5^b
Let m = a - b
Then 3^m = 5^m × k (where k is some constant)
Since 3 and 5 are prime numbers, m must be 0
Thus, a = b
Now, using the same method as in Case I, we get:
ab - c(2a - b) = -2bc
Therefore, the value of ab - c(2a - b) reduces to -2bc, irrespective of whether a = b or not.
Given, 3^a = 5^b = (75)^c
We know that 75 = 3^1 × 5^2
∴ (75)^c = (3^1 × 5^2)^c = 3^c × 5^(2c)
So, we can write 3^a = 5^b = 3^c × 5^(2c)
Let us take two cases:
Case I: When a = b
If a = b, then 3^a = 5^b
∴ 3^a = 3^c × 5^(2c) and 5^b = 3^c × 5^(2c)
∴ 3^a = 5^b
⇒ 3^a/5^b = 1
⇒ 3^(a-b) = 1
⇒ a - b = 0
Now, ab - c(2a - b) = ab - 2ac + bc
⇒ b(a - 2c) + ac
⇒ b(a - 2c) + b(a - b)
⇒ b(a - b - 2c)
Since a = b, the above expression becomes:
⇒ b(a - a - 2c) = -2bc
Therefore, ab - c(2a - b) reduces to -2bc.
Case II: When a ≠ b
If a ≠ b, then 3^a ≠ 5^b
Let m = a - b
Then 3^m = 5^m × k (where k is some constant)
Since 3 and 5 are prime numbers, m must be 0
Thus, a = b
Now, using the same method as in Case I, we get:
ab - c(2a - b) = -2bc
Therefore, the value of ab - c(2a - b) reduces to -2bc, irrespective of whether a = b or not.
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If 3 ^ a = 5 ^ b = (75) ^ c then the value of ab - c(2a b) reduces t...
If 3 ^ a = 5 ^ b = (75) ^ c then the value of ab - c(2a b) reduces to?
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If 3 ^ a = 5 ^ b = (75) ^ c then the value of ab - c(2a b) reduces to?
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If 3 ^ a = 5 ^ b = (75) ^ c then the value of ab - c(2a b) reduces to? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about If 3 ^ a = 5 ^ b = (75) ^ c then the value of ab - c(2a b) reduces to? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 3 ^ a = 5 ^ b = (75) ^ c then the value of ab - c(2a b) reduces to?.
If 3 ^ a = 5 ^ b = (75) ^ c then the value of ab - c(2a b) reduces to? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about If 3 ^ a = 5 ^ b = (75) ^ c then the value of ab - c(2a b) reduces to? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 3 ^ a = 5 ^ b = (75) ^ c then the value of ab - c(2a b) reduces to?.
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