Question

Find the value of the following expression: (3^8•2^-5•9^0)^-2•(2^-2/3^3)^4•3^28 Write your answer in simplified form. ​

Answer 1

4

Explanation:

Given expression:

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

Any number to the power of zero is 1:

`\implies (3^(8) \cdot 2^(-5) \cdot 1)^(-2) \cdot \left((2^(-2))/(3^(3))\right)^(4) \cdot 3^(28)`

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

`\textsf{Apply the exponent rule} \quad (a^b \cdot c^d)^p=a^(bp)\cdot c^(dp):`

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

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

`\textsf{Apply the exponent rule} \quad \left((a)/(b)\right)^c=(a^c)/(b^c):`

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

`\textsf{Apply the exponent rule} \quad (a^b)^c=a^(bc):`

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

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

`\textsf{Apply the exponent rule} \quad (1)/(a^n)=a^(-n)`

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

Gather like terms:

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

`\textsf{Apply the exponent rule} \quad a^b \cdot a^c=a^(b+c):`

`\implies 2^((10-8))\cdot3^((-16-12+28))`

`\implies 2^(2)\cdot3^(0)`

Any number to the power of zero is 1:

`\implies 2^(2)\cdot 1`

`\implies 2^2`

Therefore, the solution is:

`\implies 2^2=2 * 2=4`