Question

Find the value of the following expression:

(3^8 ⋅ 2−5 ⋅ 9^0)^ −2 ⋅ 2 to the power of negative 2 over 3 to the power of 3, whole to the power of 4 ⋅ 3^28 (5 points)

Write your answer in simplified form. Show all of your steps. (5 points)

Answer 1

`2^2=4`

Explanation:

`(3^8 \cdot 2^(-5) \cdot 9^0)^(-2) \cdot \left((2^(-2))/(3^3)\right)^4 \cdot3^(28)`

Using exponent rule
`a^0=1`

`\implies (3^8 \cdot 2^(-5) )^(-2) \cdot \left((2^(-2))/(3^3)\right)^4 \cdot3^(28)`

Using exponent rule
`(a^b \cdot a^c)^d=(a^(bd) \cdot a^(cd))`

`\implies 3^((8*-2)) \cdot 2^((-5*-2)) \cdot \left((2^(-2))/(3^3)\right)^4 \cdot3^(28)`

`\implies 3^(-16) \cdot 2^(10) \cdot \left((2^(-2))/(3^3)\right)^4 \cdot3^(28)`

Using exponent rule
`\left((a^b)/(a^c)\right)^d=\left((a^(bd))/(a^(cd))\right)`

`\implies 3^(-16) \cdot 2^(10) \cdot \left((2^((-2*4)))/(3^((3*4)))\right) \cdot3^(28)`

`\implies 3^(-16) \cdot 2^(10) \cdot \left((2^(-8))/(3^(12))\right) \cdot3^(28)`

Rewrite as one fraction:

`\implies (3^(-16) \cdot 2^(10) \cdot2^(-8)\cdot3^(28))/(3^(12))`

Using exponent rule
`a^b \cdot a^c=a^(b+c)`

`\implies (3^((-16+28)) \cdot 2^((10-8)))/(3^(12))`

`\implies (3^(12) \cdot 2^(2))/(3^(12))`

Cancel the common factor
`3^(12)`

`\implies 2^2`

`\implies 4`