Question

simplify the expression using the properties of exponents.\\(\frac{(3^{4})^{5}cdot 4^{3}cdot 4^{2}}{(3cdot 4)^{5}cdot 4^{3}}\\)\
show your work here\
hint: to add an exponent (\\(x^{n}\\)), type \exponent\ or press \^\

Explanation:

Step1: Simplify numerator exponents

For base 3:

$(3^4)^5 = 3^{4 \times 5} = 3^{20}$

For base 4:

$4^3 \cdot 4^2 = 4^{3+2} = 4^5$
Numerator becomes: $3^{20} \cdot 4^5$

Step2: Simplify denominator exponents

For base 3:

$(3 \cdot 4)^5 = 3^5 \cdot 4^5$
Denominator becomes: $3^5 \cdot 4^5 \cdot 4^3 = 3^5 \cdot 4^{5+3} = 3^5 \cdot 4^8$

Step3: Cancel common bases

For base 3:

$\frac{3^{20}}{3^5} = 3^{20-5} = 3^{15}$

For base 4:

$\frac{4^5}{4^8} = 4^{5-8} = 4^{-3} = \frac{1}{4^3}$

Step4: Compute final terms

$4^3 = 64$, so $\frac{3^{15}}{64}$
$3^{15} = 14348907$
$\frac{14348907}{64} = 224201.671875$ (or leave in exponential form)

Answer:

$\frac{3^{15}}{4^3}$ or $224201.671875$