Question

1. what is the solution to the exponential function $4^{2n + 4}=1024$?
2. simplify. $\frac{-(5^{2}cdot5^{3})^{4}}{5^{5}}$

extra credit reflection (up to 3 extra credit points) - answer thoroughly

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Explanation:

Step1: Solve for n in the equation

First, rewrite 1024 as a power of 4: $1024 = 4^5$. Set equal to the left side:
$$4^{2n+4} = 4^5$$
Since the bases are equal, equate exponents:
$$2n + 4 = 5$$
Subtract 4 from both sides:
$$2n = 5 - 4 = 1$$
Divide by 2:
$$n = \frac{1}{2}$$

Step2: Simplify the exponential expression

First, simplify the numerator's inner exponents:
Add exponents in the product: $5^2 \cdot 5^3 = 5^{2+3} = 5^5$
Raise to the 4th power: $(5^5)^4 = 5^{5 \cdot 4} = 5^{20}$
Now the expression is:
$$\frac{-(5^{20})}{5^5}$$
Subtract exponents for division:
$$-5^{20-5} = -5^{15}$$

Answer:

1. $n = \frac{1}{2}$
2. $-5^{15}$