Question

Find the value of the following expression: (2^8 ⋅ 5^−5 ⋅ 19^0)^−2 ⋅ 5 to the power of negative 2 over 2 to the power of 3, whole to the power of 4 ⋅ 2^28 (5 points) Write your answer in simplified form. Show all of your steps.

Answer 1

`\large\boxed{(5^2\cdot57)/(2^(26))=(1425)/(67108864)}`

Explanation:

`\left(2^8\cdot5^(-5)\cdot19^0\right)^(-2)\cdot\left((5^(-2))/(2^3)\right)^4\cdot228\\\\\text{use}\ a^(-n)=(1)/(a^n)\ \text{and}\ a^0=1\ \text{and}\ (a^n)^m=a^(nm)\\\\=\left(2^8\cdot(1)/(5^5)\cdot1\right)^(-2)\cdot\left(((1)/(5^2))/(2^3)\right)^4\cdot228=\left((2^8)/(5^5)\right)^(-2)\cdot\left((1)/(2^35^2)\right)^4\cdot228`

`=((2^8)^(-2))/((5^5)^(-2))\cdot(1^4)/((2^3)^4(5^2)^4)\cdot228=(2^(-16))/(5^(-10))\cdot(1)/(2^(12)5^8)\cdot228\\\\\text{use}\ a^n=(1)/(a^(-n))\to(1)/(a^n)=a^(-n)\\\\=2^(-16)\cdot5^(10)\cdot2^(-12)\cdot5^(-8)\cdot228\\\\\text{use}\ a^n\cdot a^m=a^(n+m)\\\\=2^(-16+(-12))\cdot5^(10+(-8))\cdot228=2^(-28)\cdot5^2\cdot228\\\\=2^(-28)\cdot5^2\cdot4\cdot57=2^(-28)\cdot5^2\cdot2^2\cdot57=2^(-28+2)\cdot5^2\cdot57\\\\=2^(-26)\cdot5^2\cdot57=(5^2\cdot57)/(2^(26))`

`\large\boxed{(5^2\cdot57)/(2^(26))=(1425)/(67108864)}`