[tex]\frac{3x^0(2x^3t^2)^4}{(4x^7y^4)^2} = \frac{3(1)(2)^4(x^3)^4(t^2)^4}{(4)^2(x^7)^2(y^4)^2}[/tex] Since, [tex]a^0 = 1[/tex] and [tex](ab)^m=a^mb^m[/tex]
[tex]\frac{3x^0(2x^3t^2)^4}{(4x^7y^4)^2} = \frac{3(16)x^{12}t^{8}}{16x^{14}y^8}[/tex] Since, [tex](a^b)^c=a^{bc}[/tex]
[tex]\frac{3x^0(2x^3t^2)^4}{(4x^7y^4)^2} = \frac{3t^{8}}{x^{14-12}y^8}[/tex]
[tex]\frac{3x^0(2x^3t^2)^4}{(4x^7y^4)^2} = \frac{3t^{8}}{x^{2}y^8}[/tex]
Thus,
[tex]\frac{3x^0(2x^3t^2)^4}{(4x^7y^4)^2} = \frac{3t^{8}}{x^{2}y^8}[/tex]
Given problem is
[tex]\frac{3x^0(2x^3t^2)^4}{(4x^7y^4)^2}[/tex]
distribute outer exponents using formula: [tex](a^mb^n)^c=a^{mc}b^{nc}[/tex], we get:
[tex]=\frac{3x^0(2^4x^{3\cdot4}t^{2\cdot4})}{4^2x^{7\cdot2}y^{4\cdot2}}[/tex]
Simplify exponents:
[tex]=\frac{3x^0(2^4x^{12}t^8)}{4^2x^{14}y^8}[/tex]
plug [tex]x^0=1[/tex]
[tex]=\frac{3\cdot1(2^4x^{12}t^8)}{4^2x^{14}y^8}[/tex]
simplify exponents
[tex]=\frac{3(16x^{12}t^8)}{16x^{14}y^8}[/tex]
simplify (closure property)
[tex]=\frac{48x^{12}t^8}{16x^{14}y^8}[/tex]
simplify exponent part using formula : [tex]\frac{a^m}{a^n}=a^{\left(m-n\right)}[/tex] we get:
[tex]=\frac{3x^{\left(12-14\right)}t^8}{y^8}[/tex]
Simplify exponents:
[tex]=\frac{3x^{\left(-2\right)}t^8}{y^8}[/tex]
send term to denominator to avoid negative exponent
[tex]=\frac{3t^8}{x^2y^8}[/tex]
Hence final answer is [tex]\frac{3t^8}{x^2y^8}[/tex].
