(10^k)/5 is a correct solution, though as it's not listed among the answer choices then we should manipulate with given expression in another way.

Which of the following is equal to 2^k*5^(k-1)?
A. 2*10^(k-1)
B. 5*10^(k-1)
C. 10^k
D. 2*10^k )
E. 10^(2k-1)

\(2^k*5^{k-1}=(2*2^{k-1})*5^{k-1}=2*10^{k-1}\).

Answer: A.

99,999^2 - 1^2 =
A. 10^10 - 2
B. (10^5 – 2)^2
C. 10^4(10^5 – 2)
D. 10^5(10^4 – 2)
E. 10^5(10^5 – 2)

Apply \(a^2-b^2=(a+b)(a-b)\):

\(99,999^2-1^2 =(99,999+1)(99,999-1)=10^5(10^5-2)\)

Answer: E.

Hope it's clear.
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= (2^k) {5^(k − 1)}
= (2^k) (5^k) / 5
= 10^k/5

none???????? seems the answeres have typo.

For second question Ans. should be (10^k)/5

Further help appreciated.
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Operations involving the same bases:
Keep the base, add or subtract the exponent (add for multiplication, subtract for division)
\(a^n*a^m=a^{n+m}\)

\(\frac{a^n}{a^m}=a^{n-m}\)

So, \(2*2^{k-1}=2^1*2^{k-1}=2^{1+k-1}=2^k\).

For more on exponents check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it helps.
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