Question

Find the value of 'a' and 'b' if:

`(sqrt243)^"a" ÷ 3^("b" + 1)` = 1 and `27^"b" - 81^(4 -"a"/2)` = 0

Answer 1

`(sqrt243)^"a" ÷ 3^("b" + 1)` = 1 and `27^"b" - 81^(4 -"a"/2)` = 0

⇒ `(sqrt(3^5))^"a" ÷  3^("b" + 1) and (3^3)^"b" - (3^4)^(4 - "a"/2)` = 0

⇒ `(3^5)^("a"/2) ÷ 3^("b" + 1) = 1 and 3^(3"b") - (3^4)^(4 - "a"/2)` = 0

⇒ `3^(((5"a")/2)) ÷  3^("b" + 1) = 1 and 3^((3"b")) - 3^(4(4 - "a"/2)` = 0

⇒ `3^(((5"a")/2 - "b" - 1)) = 1 and 3^((3"b")) - 3^(16 - 2"a")` = 0

⇒ `3^(((5"a")/2 - "b" - 1)) = 3^° and 3^(3"b") = 3^(16 - 2"a")`

⇒ `(5"a")/(2) - "b" - 1 = 0 and 3"b"` = 16 - 2a

⇒ `(5"a")/(2) - "b" = 1 and 2"a" + 3"b"` = 16

⇒ 5a - 2b = 2 and 2a + 3b = 16
Multiply the equations by 3 and 2 respectively.
⇒ 15a - 6b = 6 and 4a + 6b = 32
Adding the equations,
19a = 38
⇒ a = 2
Substitute the value of ain 5a - 2b = 2 to find b.
5a - 2b = 2
⇒ 5(2) - 2b = 2
⇒ 10 - 2b = 2
⇒ b = 4
Hence, a = 2 and b = 4.