Question

Determine whether each equation is True or False. In case you find a "False" equation, explain why is False.​

Answer 1

(1) TRUE.

(2) FALSE.

(3) FALSE.

(4) TRUE.

(5) FALSE.

Explanation:

(1)
`√(32) = 2^{(5)/(2) }`

`2^{(5)/(2) } = (√(2) )^5 = (√(2) \ * \ √(2) \ * \ √(2) \ * \ √(2) \ * \ √(2)) = 4√(2)\\\\√(32) = √(16 \ * \ 2)\ = \ √(16) \ * \ √(2) \ = \ 4√(2)`

Thus, the equation is TRUE.

(2)
`16^{(3)/(8) } = 8^2`

`16^{(3)/(8) } =(2^4)^{(3)/(8) } = 2^(3)/(2) }= (√(2) )^3 = (√(2) \ * \ √(2) \ * \ √(2)) = 2√(2) \\\\8^2 = 64`

Thus, the equation is FALSE.

(3)
`4^{(1)/(2) } = \sqrt[4]{64}`

`4^{(1)/(2) }= √(4) = 2\\\\\sqrt[4]{64} = (64)^{(1)/(4) } = (2^6)^{(1)/(4) }= 2^{(6)/(4) } = 2^{(3)/(2) }=(√(2) )^3 = (√(2) * √(2) * √(2) ) = 2√(2)`

Thus, the equation is FALSE.

(4)
`2^8 = (\sqrt[3]{16} )^6`

`2^8 = 256\\\\ (\sqrt[3]{16} )^6 = (16)^{(6)/(3) } = (2^4)^{(6)/(3) } = (2)^{(24)/(3) } = 2^8 = 256`

Thus, the equation is TRUE.

(5)
`(√(64) )^{(1)/(3) } = 8^{(1)/(6) }\\\\`

`8^{(1)/(6) } = (2^3)^{(1)/(6) } = 2^{(3)/(6) } = 2^{(1)/(2) } = √(2) \\\\(√(64) )^{(1)/(3) } = (2^6)^{(1)/(3) } = 2^{(6)/(3) } = 2^2 = 4`

Thus, the equation is FALSE.