Answer:
Simplifying the expression [tex](6^{-2})(3^{-3})(3*6)^4[/tex] we get [tex]\mathbf{108}[/tex]
Step-by-step explanation:
We need to simplify the expression [tex](6^{-2})(3^{-3})(3*6)^4[/tex]
Solving:
[tex](6^{-2})(3^{-3})(3*6)^4[/tex]
Applying exponent rule: [tex]a^{-m}=\frac{1}{a^m}[/tex]
[tex]=\frac{1}{(6^{2})}\frac{1}{(3^{3})}(18)^4\\=\frac{(18)^4}{6^{2}\:.\:3^{3}} \\[/tex]
Factors of [tex]18=2\times 3\times 3=2\times3^2[/tex]
Factors of [tex]6=2\times 3[/tex]
Replacing terms with factors
[tex]=\frac{(2\times3^2)^4}{(2\times 3)^{2}\:.\:3^{3}} \\=\frac{(2)^4\times(3^2)^4}{(2)^2\times (3)^{2}\:.\:3^{3}} \\[/tex]
Using exponent rule: [tex](a^m)^n=a^{m\times n}[/tex]
[tex]=\frac{(2)^4\times(3)^8}{(2)^2\times (3)^{2}\:.\:3^{3}} \\=\frac{2^4\times 3^8}{2^2\times 3^{2}\:.\:3^{3}}[/tex]
Using exponent rule: [tex]a^m.a^n=a^{m+n}[/tex]
[tex]=\frac{2^4\times 3^8}{2^2\times 3^{2+3}}\\=\frac{2^4\times 3^8}{2^2\times 3^{5}}[/tex]
Now using exponent rule: [tex]\frac{a^m}{a^n}=a^{m-n}[/tex]
[tex]=2^{4-2}\times 3^{8-5}\\=2^{2}\times 3^{3}\\=4\times 27\\=108[/tex]
So, simplifying the expression [tex](6^{-2})(3^{-3})(3*6)^4[/tex] we get [tex]\mathbf{108}[/tex]
