Write 14^6 in expanded form. How many factors are there?
14^6= 2^6+7^6 is expanded form.
(6+1)x(6+1)=49
Is this correct?
14^6 = 2^6 * 7^6
No addition involved; everything is being multiplied
And yes, there are 49 factors.
Oh, I'm sorry but that's not the correct expanded form for 14^6. Let's break it down:
14^6 = (2 * 7)^6 = 2^6 * 7^6
And to calculate the expanded form, we multiply:
2^6 * 7^6 = (2 * 2 * 2 * 2 * 2 * 2) * (7 * 7 * 7 * 7 * 7 * 7)
So the correct expanded form is 64 * 117,649.
Now, let's find the number of factors! To do that, we count the number of ways we can multiply the prime factors, plus 1 for the exponent of each prime:
For 2, we have 6 choices (0, 1, 2, 3, 4, 5, 6)
For 7, we also have 6 choices (0, 1, 2, 3, 4, 5, 6)
So, the total number of factors is (6 + 1) * (6 + 1) = 7 * 7 = 49 factors.
So, your calculation for the number of factors is correct, just the expanded form was a bit off. Keep up the good work!
No, that is not correct. To expand 14^6, we need to multiply 14 by itself six times:
14^6 = 14 × 14 × 14 × 14 × 14 × 14
This can be simplified as:
14^6 = 752,953,424
So, the expanded form of 14^6 is 752,953,424.
As for the number of factors, we can determine it by factoring 14^6. The prime factorization of 14 is 2 × 7, so the prime factorization of 14^6 is (2 × 7)^6. The exponents in the prime factorization determine the factors:
2^6 × 7^6 = (2 × 2 × 2 × 2 × 2 × 2) × (7 × 7 × 7 × 7 × 7 × 7)
To find the number of factors, we need to count the combinations of different exponents:
In the case of 2, we have 6 options (0, 1, 2, 3, 4, 5, 6).
In the case of 7, we also have 6 options (0, 1, 2, 3, 4, 5, 6).
Thus, the total number of factors is the product of the number of options for each prime factor:
(6 + 1) × (6 + 1) = 7 × 7 = 49
So, there are 49 factors for 14^6.
No, the expanded form of 14^6 is not 2^6 + 7^6.
To find the expanded form, you need to multiply the base (14) by itself 6 times. So, it would be:
14^6 = 14 * 14 * 14 * 14 * 14 * 14
= 752,953,6
Now, let's count the factors:
To find the number of factors of a given number, you need to prime factorize it. The prime factorization of 752,953,6 is:
752,953,6 = 2^6 * 7^6
For a number in the form a^m * b^n, the number of factors is given by (m + 1) * (n + 1).
So, in this case, there are (6 + 1) * (6 + 1) = 49 factors.
Therefore, the correct answer is:
14^6 in expanded form is 752,953,6, and it has 49 factors.