322222222222222222222222222222 * 99999999999999999999999999999999 =

Set up problem: Set up the multiplication problem. We need to multiply two very large numbers: 322222222222222222222222222222 and 99999999999999999999999999999999.Multiply numbers: Multiply the numbers. When multiplying any number by 99999999999999999999999999999999, we can use a shortcut. Multiplying by this number is almost the same as multiplying by 10^n (where n is the number of 9's) and then subtracting the original number. However, since the number of 9's is equal to the number of digits in the first number, we can simplify the process by writing down the first number (322222222222222222222222222222) and then subtracting it from a number that is the same as the first number but with each digit increased by 1 (i.e., 433333333333333333333333333333).Perform subtraction: Perform the subtraction. 433333333333333333333333333333 - 322222222222222222222222222222 = 111111111111111111111111111111Verify result: Verify the result. The subtraction seems correct, and the pattern of the result (repeating 1's) matches the expected pattern when subtracting a number with repeating 2's from a number with repeating 3's.

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$322222222222222222222222222222 \times 99999999999999999999999999999999$ =

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Math Problems

Precalculus

Write a repeating decimal as a fraction

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322222222222222222222222222222 \times 99999999999999999999999999999999

Set up problem: Set up the multiplication problem. $\newline$ We need to multiply two very large numbers: $322222222222222222222222222222$ and $99999999999999999999999999999999$ .
Multiply numbers: Multiply the numbers. $\newline$ When multiplying any number by $99999999999999999999999999999999$ , we can use a shortcut. Multiplying by this number is almost the same as multiplying by $10^n$ (where $n$ is the number of $9$ 's) and then subtracting the original number. However, since the number of $9$ 's is equal to the number of digits in the first number, we can simplify the process by writing down the first number ( $322222222222222222222222222222$ ) and then subtracting it from a number that is the same as the first number but with each digit increased by $1$ (i.e., $433333333333333333333333333333$ ).
Perform subtraction: Perform the subtraction. $433333333333333333333333333333 - 322222222222222222222222222222 = 111111111111111111111111111111$
Verify result: Verify the result. $\newline$ The subtraction seems correct, and the pattern of the result (repeating $1$ 's) matches the expected pattern when subtracting a number with repeating $2$ 's from a number with repeating $3$ 's.