Depth · one concept from first principles · step 2 of 4
Pair 2s with 5s to make powers of ten
The number is a natural number with digits. Find the value of .
Expressions
Your answer
How to solve
Strategy Organize Information in More Ways — Each 2×5 makes a 10. Pair up as many 2s with 5s as possible, leaving a small leftover number times a power of 10.
1STEP 1
Split off the extra 5s
There are 19 twos and 22 fives, so set aside 3 fives.
5²²=5¹⁹×5³
19+3=22.
8.EE.A.1Identify Subproblems2STEP 2
Pair twos with fives
Powers with the same exponent multiply into one power of the product.
2¹⁹×5¹⁹=(2×5)¹⁹=10¹⁹
Each pair is a factor of 10.
8.EE.A.1Organize Information In More Ways.
Why?
is nineteen s and nineteen s all multiplied together.
🧱An exponent counts repeated multiplicationAn exponent records only how many times a factor is used, so multiplying two powers of the same base just adds those counts together.
Why?
The factors can be reordered and regrouped into nineteen pairs without changing the product.
Why?
Changing the order of factors does not change a product.
🧱Commutativity of multiplicationa rows of b and b rows of a count the same array of dots — rows and columns just swap.
Why?
Choosing which factors to multiply first does not change a product.
🧱Associativity of multiplicationMultiplying three numbers, grouping any two first gives the same product.
3STEP 3
Write as a number times a power of 10
The leftover is 5³=125.
2¹⁹×5²²=125×10¹⁹
5³=125.
8.EE.A.3Organize Information In More Ways4STEP 4
Count the digits
Multiplying 125 by 10¹⁹ writes 19 zeros after it: 125×10²=12500 has 3+2 digits, and in the same way this number has 3+19 digits.
n=3+19=22
Each factor of 10 adds exactly one digit.
8.EE.A.3Solve An Easier Related ProblemAnswer
22
In scientific notation the number is 1.25×10²¹, which lies between 10²¹ (the smallest 22-digit number) and 10²². So it has 22 digits.
Takeaway
Every 2 paired with a 5 is a 10 -- count the tens and you've counted the zeros.
- Split off the extra 5s
- Pair twos with fives
- Write as a number times a power of 10
- Count the digits
Where next?
Another one like thissuggested
Step back and solidifysuggested
A parent dashboard for the family lives at sensimlab.com.