Question

Solve the following and explain your steps. Leave your answer in base-exponent form. (3^-2*4^-5*5^0)^-3*(4^-4/3^3)*3^3 please step by step!!!!

Answer 1

`\boxed{2^{(802)/(27)} \cdot 3^9}`

Explanation:

I will try to give as many details as possible.

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`$(3^(-2) \cdot 4^(-5) \cdot 5^0)^(-3) \cdot (4^{-(4)/(3^3) })\cdot 3^3$`

Note that

`\boxed{a^(-b) = (1)/(a^b), a\\eq 0 }`

The denominator can't be 0 because it would be undefined.

So, we can solve the expression inside both parentheses.

`\left((1)/(3^2) \cdot (1)/(4^5) \cdot 5^0 \right)^(-3) \cdot \left(\frac{1}{4^{(4)/(3^3) } }\right)\cdot 3^3`

Also,

`\boxed{a^(0) = 1, a\\eq 0 }`

`\left((1)/(9) \cdot (1)/(1024) \cdot 1 \right)^(-3) \cdot \left(\frac{1}{4^{(4)/(27) } }\right)\cdot 27`

Note

`\boxed{(1)/(a) \cdot (1)/(b)= (1)/(ab) , a, b \\eq 0}`

`\left((1)/(9216) \right)^(-3) \cdot \left(\frac{1}{4^{(4)/(27) } }\right)\cdot 27`

`\left((1)/(9216) \right)^(-3) \cdot \left(\frac{27}{4^{(4)/(27) } }\right)`

`\left( (1)/(\left((1)/(9216)\right)^3) \right)\cdot \left(\frac{27}{4^{(4)/(9) } }\right)`

`\left( (1)/(\left((1)/(9216)\right)^3) \right)\cdot \left(\frac{27}{4^{(4)/(27) } }\right)`

Note

`\boxed{(1)/((1)/(a) ) = a}`

`9216^3\cdot \left(\frac{27}{4^{(4)/(9) } }\right)`

`\left(\frac{ 9216^3\cdot 27}{4^{(4)/(27) } }\right)`

Once

`9216=2^(10)\cdot 3^2 \implies 9216^3=2^(30)\cdot 3^6`

`\boxed{(a \cdot b)^n=a^n \cdot b^n}`

And

`$4^{(4)/(27)} = 2^{(8)/(27) $`

We have

`\left(\frac{ 2^(30) \cdot 3^6\cdot 27}{2^{(8)/(27) } }\right)`

Also, once

`\boxed{(c^a)/(c^b)=c^(a-b)}`

`2^{30-(8)/(27)} \cdot 3^6\cdot 27`

As

`30-(8)/(27) = (30 \cdot 27)/(27)-(8)/(27) =(802)/(27)`

`2^{30-(8)/(27)} \cdot 3^6\cdot 27 = 2^{(802)/(27)} \cdot 3^6 \cdot 3^3`

`2^{(802)/(27)} \cdot 3^9`