3^-3 * 2^-3 *6^3 / (4^0)^2
= 6^-3 * 6^3 / (1)^2
= 1 / 1
= 1
---------------------------------------
2^4 * 3^5 / (2*3)^5
= 2^4 * 3^5 / 2^5 *3^5
= 1/ 2^1
= 1/2
---------------------------
(3 * 2)^4 * 3^-3 / 2^3 * 3
=3^4 * 2^4 * 3^-3 / 2^3 * 3
= 3^1 * 2^4 / 2^3 * 3
= 2^1
= 2
-----------------------
3^2 * 4^3 * 2^-1 / (3 * 4)^2
= 3^2 * 4^3 / 3^2 * 4^2 * 2^1
= 4^1 / 2^1
= 4/2
= 2
Exponential expressions can be represented in form of indices.
The results of the expressions are:
[tex]\mathbf{(a)\ \frac{3^{-3} \cdot 2^{-3} \cdot 6^3}{(4^0)^2}}[/tex]
Express 4^0 as 1
[tex]\mathbf{ \frac{3^{-3} \cdot 2^{-3} \cdot 6^3}{(4^0)^2} = \frac{3^{-3} \cdot 2^{-3} \cdot 6^3}{1^2}}[/tex]
Express 1^2 as 1
[tex]\mathbf{ \frac{3^{-3} \cdot 2^{-3} \cdot 6^3}{(4^0)^2} = 3^{-3} \cdot 2^{-3} \cdot 6^3}[/tex]
Express 6 as 2 * 3
[tex]\mathbf{ \frac{3^{-3} \cdot 2^{-3} \cdot 6^3}{(4^0)^2} = 3^{-3} \cdot 2^{-3} \cdot (2 \times 3)^3}[/tex]
Expand
[tex]\mathbf{ \frac{3^{-3} \cdot 2^{-3} \cdot 6^3}{(4^0)^2} = 3^{-3} \cdot 2^{-3} \cdot 2^3 \times 3^3}[/tex]
Rewrite as:
[tex]\mathbf{ \frac{3^{-3} \cdot 2^{-3} \cdot 6^3}{(4^0)^2} = 3^{-3} \times 3^3 \cdot 2^{-3} \times 2^3 }[/tex]
Apply law of indices
[tex]\mathbf{ \frac{3^{-3} \cdot 2^{-3} \cdot 6^3}{(4^0)^2} = 3^{-3+3} \cdot 2^{-3+3} }[/tex]
[tex]\mathbf{ \frac{3^{-3} \cdot 2^{-3} \cdot 6^3}{(4^0)^2} = 3^{0} \cdot 2^0 }[/tex]
[tex]\mathbf{ \frac{3^{-3} \cdot 2^{-3} \cdot 6^3}{(4^0)^2} = 1 \cdot 1 }[/tex]
[tex]\mathbf{ \frac{3^{-3} \cdot 2^{-3} \cdot 6^3}{(4^0)^2} = 1}[/tex]
[tex]\mathbf{(b)\ \frac{2^4 \cdot 3^5}{(2\cdot 3)^5}}[/tex]
Open bracket
[tex]\mathbf{\frac{2^4 \cdot 3^5}{(2\cdot 3)^5} = \frac{2^4 \cdot 3^5}{2^5 \cdot 3^5}}[/tex]
Apply laws of indices
[tex]\mathbf{\frac{2^4 \cdot 3^5}{(2\cdot 3)^5} = 2^{4-5} \cdot 3^{5-5}}[/tex]
[tex]\mathbf{\frac{2^4 \cdot 3^5}{(2\cdot 3)^5} = 2^{-1} \cdot 3^{0}}[/tex]
[tex]\mathbf{\frac{2^4 \cdot 3^5}{(2\cdot 3)^5} = 2^{-1} \cdot 1}[/tex]
[tex]\mathbf{\frac{2^4 \cdot 3^5}{(2\cdot 3)^5} = 2^{-1}}[/tex]
Rewrite as:
[tex]\mathbf{\frac{2^4 \cdot 3^5}{(2\cdot 3)^5} = \frac{1}{2}}[/tex]
[tex]\mathbf{(c)\ \frac{(3 \cdot 2)^4 \cdot 3^{-3}}{2^3 \cdot 3}}[/tex]
Expand bracket
[tex]\mathbf{\frac{(3 \cdot 2)^4 \cdot 3^{-3}}{2^3 \cdot 3} = \frac{3^4 \cdot 2^4 \cdot 3^{-3}}{2^3 \cdot 3}}[/tex]
Apply law of indices
[tex]\mathbf{\frac{(3 \cdot 2)^4 \cdot 3^{-3}}{2^3 \cdot 3} = 3^{4-3-1} \cdot 2^{4-3}}[/tex]
[tex]\mathbf{\frac{(3 \cdot 2)^4 \cdot 3^{-3}}{2^3 \cdot 3} = 3^{0} \cdot 2^{1}}[/tex]
[tex]\mathbf{\frac{(3 \cdot 2)^4 \cdot 3^{-3}}{2^3 \cdot 3} = 1 \cdot 2}[/tex]
[tex]\mathbf{\frac{(3 \cdot 2)^4 \cdot 3^{-3}}{2^3 \cdot 3} = 2}[/tex]
[tex]\mathbf{(d)\ \frac{3^2 \cdot 4^3 \cdot 2^{-1}}{(3\cdot 4)^2}}[/tex]
Express 4 as 2^2
[tex]\mathbf{\frac{3^2 \cdot 4^3 \cdot 2^{-1}}{(3\cdot 4)^2} = \frac{3^2 \cdot (2^2)^3 \cdot 2^{-1}}{(3\cdot 2^2)^2}}[/tex]
[tex]\mathbf{\frac{3^2 \cdot 4^3 \cdot 2^{-1}}{(3\cdot 4)^2} = \frac{3^2 \cdot 2^6 \cdot 2^{-1}}{(3\cdot 2^2)^2}}[/tex]
Open bracket
[tex]\mathbf{\frac{3^2 \cdot 4^3 \cdot 2^{-1}}{(3\cdot 4)^2} = \frac{3^2 \cdot 2^6 \cdot 2^{-1}}{3^2\cdot 2^4}}[/tex]
Apply law of indices
[tex]\mathbf{\frac{3^2 \cdot 4^3 \cdot 2^{-1}}{(3\cdot 4)^2} = 3^{2-2} \cdot 2^{6-1-4}}[/tex]
[tex]\mathbf{\frac{3^2 \cdot 4^3 \cdot 2^{-1}}{(3\cdot 4)^2} = 3^{0} \cdot 2^{1}}[/tex]
[tex]\mathbf{\frac{3^2 \cdot 4^3 \cdot 2^{-1}}{(3\cdot 4)^2} = 1 \cdot 2}[/tex]
[tex]\mathbf{\frac{3^2 \cdot 4^3 \cdot 2^{-1}}{(3\cdot 4)^2} = 2}[/tex]
Read more about indices and exponential expressions at:
https://brainly.com/question/12916986
