Question

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

Explanation:

Step1: Simplify numerator exponents

First, combine the base-5 terms in the numerator using $a^m \cdot a^n = a^{m+n}$, and apply $(a^m)^n = a^{m \cdot n}$ to the base-4 term:
Numerator: $5^{2+2} \cdot 4^{3 \cdot 5} = 5^4 \cdot 4^{15}$
Denominator: Keep as is for now: $5^2 \cdot (5 \cdot 4)^5$

Step2: Expand denominator product

Use $(a \cdot b)^n = a^n \cdot b^n$ on the denominator's $(5 \cdot 4)^5$ term:
Denominator: $5^2 \cdot 5^5 \cdot 4^5$

Step3: Combine denominator base-5 terms

Combine the base-5 terms in the denominator using $a^m \cdot a^n = a^{m+n}$:
Denominator: $5^{2+5} \cdot 4^5 = 5^7 \cdot 4^5$

Step4: Simplify base-5 terms

Use $\frac{a^m}{a^n} = a^{m-n}$ for the base-5 terms:
$\frac{5^4}{5^7} = 5^{4-7} = 5^{-3} = \frac{1}{5^3}$

Step5: Simplify base-4 terms

Use $\frac{a^m}{a^n} = a^{m-n}$ for the base-4 terms:
$\frac{4^{15}}{4^5} = 4^{15-5} = 4^{10}$

Step6: Combine remaining terms

Multiply the simplified base-5 and base-4 results:
$\frac{4^{10}}{5^3}$
We can also calculate the numerical values: $4^{10}=1048576$, $5^3=125$, so $\frac{1048576}{125}=8388.608$

Answer:

$\frac{4^{10}}{5^3}$ or $8388.608$