How many factors of 10 are in each power? Number of factors of 10 in 10^(2) : ? Number of factors of 10 in 10^(3) : ? 10^(2)=1010 10^(3)=1010*10
The Correct Answer and Explanation is:
To understand the number of factors of 10 in each power of 10, let’s first break down the powers of 10 and their components.
Understanding Powers of 10:
- ( 10^2 ) means ( 10 \times 10 ), which is ( 100 ).
- ( 10^3 ) means ( 10 \times 10 \times 10 ), which is ( 1000 ).
Breaking Down Factors of 10:
A factor of 10 can be expressed as the product of 10, which itself can be written as:
[
10 = 2 \times 5
]
Thus, every factor of 10 contains both the prime factors 2 and 5. Now, we need to figure out how many times 10 divides into a given number.
Number of Factors of 10 in ( 10^2 ):
We start with the expression for ( 10^2 ):
[
10^2 = 10 \times 10 = 100
]
Since ( 100 = 2^2 \times 5^2 ), it contains exactly two 10s (i.e., two factors of 10). This is because each 10 can be broken into one factor of 2 and one factor of 5. So, in ( 10^2 ), there are exactly 2 factors of 10.
Number of Factors of 10 in ( 10^3 ):
Now, let’s consider ( 10^3 ):
[
10^3 = 10 \times 10 \times 10 = 1000
]
Since ( 1000 = 2^3 \times 5^3 ), it contains exactly three 10s (i.e., three factors of 10). This means that in ( 10^3 ), there are exactly 3 factors of 10.
Conclusion:
- The number of factors of 10 in ( 10^2 ) is 2.
- The number of factors of 10 in ( 10^3 ) is 3.
This relationship holds generally: the number of factors of 10 in ( 10^n ) is exactly n because ( 10^n ) is the product of ( n ) tens, and each of those tens contributes one factor of 10. Therefore, ( 10^n ) contains ( n ) factors of 10.