If the expression 410 was simplified using the Power Rule of Exponents, what is an equivalent expanded expression?
a) (4 ^ 12) ^ 2
b) (4 ^ 5) ^ 2
c) 4 * 4 * 4 * 4 * 4 * 4
d) (4 ^ 2) ^ 8
The Power Rule of Exponents states that (a^b)^c is equal to a^(b*c).
In this case, the expression 410 is already in its simplified form (there are no exponents or base). Therefore, there is no equivalent expanded expression using the Power Rule of Exponents.
The correct answer is: There is no equivalent expanded expression.
pick an answer
In that case, option (c) 4 * 4 * 4 * 4 * 4 * 4 is the closest equivalent expanded expression.
To simplify the expression 410 using the Power Rule of Exponents, we need to decompose the number into its prime factors. The prime factorization of 410 is 2 * 5 * 41.
Using the Power Rule of Exponents, we can write the expression as 2^1 * 5^1 * 41^1.
Therefore, the equivalent expanded expression is c) 4 * 4 * 4 * 4 * 4 * 4.
To solve this question, first let's review the Power Rule of Exponents. According to the Power Rule, when you raise a number to an exponent and then raise the result to another exponent, you multiply the exponents.
Now, let's apply the Power Rule of Exponents to the given expression, 410.
Since there are no specified exponents in the expression 410, we can assume that 10 is the exponent of 4. Therefore, we can rewrite 410 as 4^10.
Now, let's look at the answer choices:
a) (4^12)^2 = 4^(12*2) = 4^24
b) (4^5)^2 = 4^(5*2) = 4^10 (not equal to 410)
c) 4*4*4*4*4*4 = 4^6 (not equal to 410)
d) (4^2)^8 = 4^(2*8) = 4^16 (not equal to 410)
From the answer choices, we can see that none of them are equivalent to 410. Therefore, the correct answer is none of the above.