Question

Part A: If (6^2)^X = 1, what is the value of x? Explain your answer. (5 points)

Part B: If (6^9)^x = 1, what are the possible values of x? Explain your answer.

Answer 1

Part A: X=0

Part B: x=0

Explanation:

Part A

(6^2)^X = 1

Applying the exponent rule:
`(a^b)^c = a^(bc)`

So, our equation will become:

`6^(2X) = 1`

We know if f(x) = g(x) then ln(f(x))= ln(g(x))

SO, taking natural logarithm ln on both sides and solving.

`ln(6^(2X)) =ln(1)`

We know,
`log(a^b) = b.loga` Applying the rule,

`2Xln6 =ln(1)\\We\,\,know\,\,ln(1)=0\\2Xln6 =0\\Solving:\\X=0`

Part B

(6^9)^x = 1

Applying the exponent rule:
`(a^b)^c = a^(bc)`

So, our equation will become:

`6^(9x) = 1`

We know if f(x) = g(x) then ln(f(x))= ln(g(x))

SO, taking natural logarithm ln on both sides and solving.

`ln(6^(9x)) =ln(1)`

We know,
`ln(a^b) = b.lna` Applying the rule,

`9xln6 =ln(1)\\We\,\,know\,\,ln(1)=0\\9xln6 =0\\Solving:\\x=0`