Question

Find the value of y such that 9^y = 3^5 * 9^6 / 27^3. Explain how you solved this problem.

Answer 1

Explanation:

`9^y=(3^5*9^6)/(27^3) \\\\\\(3^2)^y=(3^5*(3^2)^6)/((3^3)^3) \\\\\\3^(2y)=3^(5+12-9)\\\\\\2y=8\\\\\\y=4\\`

Answer 2

`{9}^(y) = \frac{ {3}^(5) * {9}^(6) }{ {27}^(3) } \\`

• express in terms of 3:

`{3}^(2y) = \frac{ {3}^(5) * {3}^(12) }{ {3}^(9) } \\`

• from law of indices:

`{a}^(n) * {a}^(m) = {a}^((n + m)) \\ \\ \frac{ {a}^(n) }{ {a}^(m) } = {a}^((n - m))`

• therefore, let's apply the laws:

`{3}^(2y) = {3}^((5 + 12 - 9)) \\ \\ {3}^(2y) = {3}^(8)`

• from third law of indices:

`\{ {a}^(x) = {a}^(y) \} \equiv \{x = y \}`

• apply the law:

`2y = 8 \\ y = (8)/(2 ) \\ \\ { \underline{ \underline{ \: \: y = 4 \: \: }}}`