RBSE Solutions Class 10 Maths Chapter 1 Vedic Mathematics Exercise 1.3

Find the square root by the Vedic Methods:

Question 1. Find the square root of 2116.
Answer: We will find the square root of 2116 using the Vedic method. The number 2116 is grouped into pairs from the right: 21 | 16. The largest perfect square less than or equal to 21 is 16 (which is \( 4^2 \)). So, the first digit of the square root is 4.
\[ \begin{array}{r|rr} 4 & 21 & 16 \\ \times 4 & 16 & \\ \hline 8 & 5 & 16 \\ 86 & & \\ \times 6 & & 516 \\ \hline & & 0 \end{array} \]
Thus, the square root of 2116 is 46.
In simple words: We find the square root of 2116 using a special Vedic method, which involves dividing the number into pairs and finding the largest square. This process gives us 46 as the answer.

🎯 Exam Tip: Always remember to group the digits in pairs starting from the right for square root calculations. The first digit of the root comes from the largest square less than the first pair of digits.

Question 2. Find the square root of 4225.
Answer: To find the square root of 4225 using the Vedic method, we first group the digits into pairs from the right: 42 | 25. The largest perfect square less than or equal to 42 is 36 (which is \( 6^2 \)). So, the first digit of the square root is 6.
\[ \begin{array}{r|rr} 6 & 42 & 25 \\ \times 6 & 36 & \\ \hline 12 & 6 & 25 \\ 125 & & \\ \times 5 & & 625 \\ \hline & & 0 \end{array} \]
Therefore, the square root of 4225 is 65.
In simple words: We use the Vedic method to find the square root of 4225. We pair numbers and find the nearest square, which leads us to the answer 65.

🎯 Exam Tip: When the number ends in 25, the square root will always end in 5, which can help verify your last digit.

Question 3. Find the square root of 6889.
Answer: We will find the square root of 6889 using the Vedic method. First, we group the digits into pairs from the right: 68 | 89. The largest perfect square less than or equal to 68 is 64 (which is \( 8^2 \)). So, the first digit of the square root is 8.
\[ \begin{array}{r|rr} 8 & 68 & 89 \\ \times 8 & 64 & \\ \hline 16 & 4 & 89 \\ 163 & & \\ \times 3 & & 489 \\ \hline & & 0 \end{array} \]
Therefore, the square root of 6889 is 83.
In simple words: We apply the Vedic method to calculate the square root of 6889. By pairing the digits and performing step-by-step division, we find that the square root is 83.

🎯 Exam Tip: Practice recognizing the last digit of squares (e.g., 9 can come from \( 3^2 \) or \( 7^2 \)) to quickly narrow down possible square roots.

Question 4. Find the square root of 59049.
Answer: To find the square root of 59049 by the Vedic method, we group the digits into pairs from the right, with the leftmost digit possibly being a single one: 5 | 90 | 49. The largest perfect square less than or equal to 5 is 4 (which is \( 2^2 \)). So, the first digit of the square root is 2.
\[ \begin{array}{r|rrr} 2 & 5 & 90 & 49 \\ \times 2 & 4 & & \\ \hline 4 & 1 & 90 & \\ 44 & & & \\ \times 4 & & 176 & \\ \hline 48 & & 14 & 49 \\ 483 & & & \\ \times 3 & & & 1449 \\ \hline & & & 0 \end{array} \]
Therefore, the square root of 59049 is 243.
In simple words: We find the square root of 59049 using the Vedic technique. We divide the number into sections and calculate each digit of the root step-by-step, resulting in 243.

🎯 Exam Tip: When the number of digits is odd, the first group will be a single digit. Remember to calculate the corrected dividend carefully in each step.

Question 5. Find the square root of 125316.
Answer: We calculate the square root of 125316 using the Vedic method. We group the digits from the right into pairs: 12 | 53 | 16. The largest perfect square less than or equal to 12 is 9 (which is \( 3^2 \)). So, the first digit of the square root is 3.
\[ \begin{array}{r|rrr} 3 & 12 & 53 & 16 \\ \times 3 & 9 & & \\ \hline 6 & 3 & 53 & \\ 65 & & & \\ \times 5 & & 325 & \\ \hline 70 & & 28 & 16 \\ 704 & & & \\ \times 4 & & & 2816 \\ \hline & & & 0 \end{array} \]
Therefore, the square root of 125316 is 354.
In simple words: Using the Vedic technique, we determine the square root of 125316. The process involves pairing numbers and finding the closest square at each stage, leading to the result 354.

🎯 Exam Tip: Keep your columns and calculations neat to avoid errors, especially with larger numbers. A small mistake in an early step can affect the entire solution.

Question 6. Find the square root of 169744.
Answer: We will find the square root of 169744 using the Vedic method. We group the digits into pairs from the right: 16 | 97 | 44. The largest perfect square less than or equal to 16 is 16 (which is \( 4^2 \)). So, the first digit of the square root is 4.
\[ \begin{array}{r|rrr} 4 & 16 & 97 & 44 \\ \times 4 & 16 & & \\ \hline 8 & 0 & 97 & \\ 81 & & & \\ \times 1 & & 81 & \\ \hline 82 & & 16 & 44 \\ 822 & & & \\ \times 2 & & & 1644 \\ \hline & & & 0 \end{array} \]
Therefore, the square root of 169744 is 412.
In simple words: We calculate the square root of 169744 using the Vedic method by pairing numbers and performing step-by-step divisions. This systematic process gives us 412 as the square root.

🎯 Exam Tip: When the divisor is larger than the dividend, or when a trial divisor multiplied by a digit leads to a result greater than the dividend, the next digit of the quotient is usually 0. Here, it leads to a pattern where the divisor changes as we include more digits from the quotient.

Question 7. Find the square root of 1265625.
Answer: We will find the square root of 1265625 using the Vedic method. We group the digits into pairs from the right, with the leftmost digit being a single one: 1 | 26 | 56 | 25. The largest perfect square less than or equal to 1 is 1 (which is \( 1^2 \)). So, the first digit of the square root is 1.
\[ \begin{array}{r|rrrr} 1 & 1 & 26 & 56 & 25 \\ \times 1 & 1 & & & \\ \hline 2 & 0 & 26 & & \\ 21 & & & & \\ \times 1 & & 21 & & \\ \hline 22 & & 5 & 56 & \\ 222 & & & & \\ \times 2 & & & 444 & \\ \hline 224 & & & 112 & 25 \\ 2245 & & & & \\ \times 5 & & & & 11225 \\ \hline & & & & 0 \end{array} \]
Therefore, the square root of 1265625 is 1125.
In simple words: We calculate the square root of 1265625 using the Vedic division method. We pair digits and find the largest square at each stage, leading us to the final answer of 1125.

🎯 Exam Tip: Numbers ending in 25 often have square roots ending in 5. This can be a useful check for your final answer.

Question 8. Find the square root of 1522756.
Answer: We will find the square root of 1522756 using the Vedic method. First, we group the digits into pairs: 1 | 52 | 27 | 56. The largest perfect square less than or equal to 1 is 1 (which is \( 1^2 \)). So, the first digit of the square root is 1.
\[ \begin{array}{r|rrrr} 1 & 1 & 52 & 27 & 56 \\ \times 1 & 1 & & & \\ \hline 2 & 0 & 52 & & \\ 22 & & & & \\ \times 2 & & 44 & & \\ \hline 24 & & 8 & 27 & \\ 243 & & & & \\ \times 3 & & & 729 & \\ \hline 246 & & & 98 & 56 \\ 2464 & & & & \\ \times 4 & & & & 9856 \\ \hline & & & & 0 \end{array} \]
Therefore, the square root of 1522756 is 1234.
In simple words: Using the Vedic technique, we find the square root of 1522756. This involves separating the number into pairs and solving for each digit of the root step-by-step, resulting in 1234.

🎯 Exam Tip: Pay close attention to the placement of the digits in the quotient and the construction of the trial divisor in each step to ensure accuracy.

Using the Vedic Methods, find the cube root:

Question 9. Find the cube root of 68921.
Answer: We will find the cube root of 68921 using the Vedic method. We group the digits into blocks of three from the right: 68 | 921. The largest perfect cube less than or equal to 68 is 64 (which is \( 4^3 \)). So, the first digit of the cube root is 4. The last digit of 68921 is 1, so the last digit of its cube root must also be 1. Thus, the cube root is 41.
The calculation steps are as follows:
\[ \begin{array}{r|rr} \text{Steps} & 41 & \\ \downarrow & 68 & 921 \\ -4^3 & 64 & \\ \hline & 4 & 921 \\ -3 \times 4^2 \times 1 & 48 & \\ \hline & 1 & 21 \\ -3 \times 4 \times 1^2 & 12 & \\ \hline & & 1 \\ -1^3 & & 1 \\ \hline & & 0 \end{array} \]
Therefore, the cube root of 68921 is 41.
In simple words: To find the cube root of 68921 using the Vedic method, we divide the number into groups of three digits. We find the cube root of the first group and use the last digit of the original number to determine the second digit, arriving at 41.

🎯 Exam Tip: For cube roots, grouping digits in threes from the right is crucial. The unit digit of the number often directly tells you the unit digit of its cube root (e.g., ends in 1 means root ends in 1; ends in 8 means root ends in 2, etc.).

Question 10. Find the cube root of 636056.
Answer: We will find the cube root of 636056 using the Vedic method. We group the digits into blocks of three from the right: 636 | 056. The largest perfect cube less than or equal to 636 is 512 (which is \( 8^3 \)). So, the first digit of the cube root is 8. The last digit of 636056 is 6, so the last digit of its cube root must also be 6. Thus, the cube root is 86.
The calculation steps are as follows:
\[ \begin{array}{r|rr} & 86 & \\ \downarrow & 636 & 056 \\ -8^3 & 512 & \\ \hline & 124 & 056 \\ -3 \times 8^2 \times 6 & 1152 & \\ \hline & & 885 \\ -3 \times 8 \times 6^2 & & 864 \\ \hline & & 216 \\ -6^3 & & 216 \\ \hline & & 0 \end{array} \]
Therefore, the cube root of 636056 is 86.
In simple words: To get the cube root of 636056, we divide the number into sections of three digits. We then calculate the cube root of the first section and find the second digit using the final digit of the original number. This process yields 86.

🎯 Exam Tip: Familiarize yourself with cubes of numbers up to 10 to quickly determine the first and last digits of a cube root, as this simplifies the Vedic method steps.

Question 11. Find the cube root of 314432.
Answer: We will find the cube root of 314432 using the Vedic method. We group the digits into blocks of three from the right: 314 | 432. The largest perfect cube less than or equal to 314 is 216 (which is \( 6^3 \)). So, the first digit of the cube root is 6. The last digit of 314432 is 2, so the last digit of its cube root must be 8. Thus, the cube root is 68.
The calculation steps are as follows:
\[ \begin{array}{r|rr} \text{Steps} & 68 & \\ \downarrow & 314 & 432 \\ -6^3 & 216 & \\ \hline & 98 & 432 \\ -3 \times 6^2 \times 8 & 864 & \\ \hline & 120 & 32 \\ -3 \times 6 \times 8^2 & 1152 & \\ \hline & & 512 \\ -8^3 & & 512 \\ \hline & & 0 \end{array} \]
Therefore, the cube root of 314432 is 68.
In simple words: To find the cube root of 314432, we use the Vedic method by splitting the number into groups of three digits. We find the cube root of the first group and then use the last digit of the original number to help find the second digit, which gives us 68.

🎯 Exam Tip: Pay attention to the relationship between the last digit of a number and the last digit of its cube root (e.g., if a number ends in 2, its cube root ends in 8, and vice versa). This can speed up determining the unit digit.

Question 12. Find the cube root of 493039.
Answer: We will find the cube root of 493039 using the Vedic method. We group the digits into blocks of three from the right: 493 | 039. The largest perfect cube less than or equal to 493 is 343 (which is \( 7^3 \)). So, the first digit of the cube root is 7. The last digit of 493039 is 9, so the last digit of its cube root must also be 9. Thus, the cube root is 79.
The calculation steps are as follows:
\[ \begin{array}{r|rr} & 79 & \\ \downarrow & 493 & 039 \\ -7^3 & 343 & \\ \hline & 150 & 039 \\ -3 \times 7^2 \times 9 & 1323 & \\ \hline & & 1773 \\ -3 \times 7 \times 9^2 & & 1701 \\ \hline & & 729 \\ -9^3 & & 729 \\ \hline & & 0 \end{array} \]
Therefore, the cube root of 493039 is 79.
In simple words: We find the cube root of 493039 by separating it into groups of three digits. We then use the largest cube for the first group and the last digit for the second, leading to a cube root of 79.

🎯 Exam Tip: When using the Vedic cube root method, precisely calculating each of the intermediate terms \( 3a^2b \), \( 3ab^2 \), and \( b^3 \) is essential for accuracy.

Question 13. Find the cube root of 8365427.
Answer: We will find the cube root of 8365427 using the Vedic method. We group the digits into blocks of three from the right: 8 | 365 | 427. The largest perfect cube less than or equal to 8 is 8 (which is \( 2^3 \)). So, the first digit of the cube root is 2. The last digit of 8365427 is 7, so the last digit of its cube root must be 3. This means the cube root is 203.
The calculation steps are as follows:
\[ \begin{array}{r|rrr} \text{Steps} & 203 & & \\ \downarrow & 8 & 365 & 427 \\ -2^3 & 8 & & \\ \hline & 0 & 365 & \\ -3 \times 2^2 \times 0 & 0 & & \\ \hline & 365 & 427 \\ -3 \times 2 \times 0^2 & 0 & & \\ \hline & 365 & 427 \\ -0^3 & 0 & & \\ \hline & 365 & 427 \\ -3 \times 20^2 \times 3 & 3600 & \\ \hline & & 542 \\ -3 \times 20 \times 3^2 & & 540 \\ \hline & & 27 \\ -3^3 & & 27 \\ \hline & & 0 \end{array} \]
Therefore, the cube root of 8365427 is 203.
In simple words: To find the cube root of 8365427, we use the Vedic method by splitting the number into groups of three digits. We find the cube root of the first group and then use the last digit of the original number to help find the second digit, which gives us 203.

🎯 Exam Tip: For three-digit cube roots, remember that the middle digit can be zero. Follow the systematic subtraction of terms like \( 3a^2b \) and \( 3ab^2 \) carefully for each part of the number.

Question 14. Find the cube root of 1061208.
Answer: We will find the cube root of 1061208 using the Vedic method. We group the digits into blocks of three from the right: 1 | 061 | 208. The largest perfect cube less than or equal to 1 is 1 (which is \( 1^3 \)). So, the first digit of the cube root is 1. The last digit of 1061208 is 8, so the last digit of its cube root must be 2. This suggests the cube root is 102.
The calculation steps are as follows:
\[ \begin{array}{r|rrr} \text{Steps} & 102 & & \\ \downarrow & 1 & 061 & 208 \\ -1^3 & 1 & & \\ \hline & 0 & 061 & \\ -3 \times 1^2 \times 0 & 0 & & \\ \hline & 61 & 208 \\ -3 \times 1 \times 0^2 & 0 & & \\ \hline & 61 & 208 \\ -0^3 & 0 & & \\ \hline & 61 & 208 \\ -3 \times 10^2 \times 2 & 600 & \\ \hline & & 1208 \\ -3 \times 10 \times 2^2 & & 120 \\ \hline & & 8 \\ -2^3 & & 8 \\ \hline & & 0 \end{array} \]
Therefore, the cube root of 1061208 is 102.
In simple words: To find the cube root of 1061208, we use the Vedic method. We divide the number into groups of three and determine the cube root of each section step-by-step. This systematic process gives us 102.

🎯 Exam Tip: When a middle digit of the cube root is zero, the intermediate terms involving that digit will also be zero, simplifying those steps in the calculation.

Question 15. Find the cube root of 8489664.
Answer: We will find the cube root of 8489664 using the Vedic method. We group the digits into blocks of three from the right: 8 | 489 | 664. The largest perfect cube less than or equal to 8 is 8 (which is \( 2^3 \)). So, the first digit of the cube root is 2. The last digit of 8489664 is 4, so the last digit of its cube root must be 4. This implies the cube root is 204.
The calculation steps are as follows:
\[ \begin{array}{r|rrr} \text{Steps} & 204 & & \\ \downarrow & 8 & 489 & 664 \\ -2^3 & 8 & & \\ \hline & 0 & 489 & \\ -3 \times 2^2 \times 0 & 0 & & \\ \hline & 489 & 664 \\ -3 \times 2 \times 0^2 & 0 & & \\ \hline & 489 & 664 \\ -0^3 & 0 & & \\ \hline & 489 & 664 \\ -3 \times 20^2 \times 4 & 4800 & \\ \hline & & 9664 \\ -3 \times 20 \times 4^2 & & 960 \\ \hline & & 64 \\ -4^3 & & 64 \\ \hline & & 0 \end{array} \]
Therefore, the cube root of 8489664 is 204.
In simple words: Using the Vedic method, we find the cube root of 8489664. We break down the number into groups of three digits, find the initial cube root, and then use the last digit to confirm and complete the step-by-step calculations, resulting in 204.

🎯 Exam Tip: Remember the cube properties for digits 1-9 (e.g., \( 2^3 = 8 \), \( 3^3 = 27 \), \( 4^3 = 64 \)) to correctly identify the first and last digits of the cube root.

Question 16. Find the cube root of 200201625.
Answer: We will find the cube root of 200201625 using the Vedic method. We group the digits into blocks of three from the right: 200 | 201 | 625. The largest perfect cube less than or equal to 200 is 125 (which is \( 5^3 \)). So, the first digit of the cube root is 5. The last digit of 200201625 is 5, so the last digit of its cube root must also be 5. This implies the cube root is 585.
The calculation steps are as follows:
\[ \begin{array}{r|rrr} & 585 & & \\ \downarrow & 200 & 201 & 625 \\ -5^3 & 125 & & \\ \hline & 75 & 201 & \\ -3 \times 5^2 \times 8 & 600 & & \\ \hline & 15 & 201 \\ -3 \times 5 \times 8^2 & 960 & \\ \hline & & 5601 \\ -8^3 & & 512 & \\ \hline & & 5089 & 625 \\ -3 \times 58^2 \times 5 & 50460 & \\ \hline & & & 43625 \\ -3 \times 58 \times 5^2 & & & 4350 \\ \hline & & & 125 \\ -5^3 & & & 125 \\ \hline & & & 0 \end{array} \]
Therefore, the cube root of 200201625 is 585.
In simple words: To get the cube root of 200201625, we use the Vedic method, grouping the digits in threes. We find the cube root of the initial number and then apply a series of specific subtractions based on the digits we find, finally reaching 585.

🎯 Exam Tip: For large numbers, break down the cube root calculation into finding each digit sequentially. Ensure that the terms \( 3a^2b \), \( 3ab^2 \), and \( b^3 \) are correctly calculated and subtracted at each stage.

Question 17. Find the cube root of 2584754853.
Answer: We will find the cube root of 2584754853 using the Vedic method. We group the digits into blocks from the right. The first digit of the cube root is 6 because \( 6^3 = 216 \). The last digit is 7 because the number ends in 3 (\( 7^3 = 343 \)). This implies the cube root is 637.
The calculation steps are as follows, treating the number as consisting of parts to find the digits 6, 3, and 7 sequentially:
\[ \begin{array}{r|rrr} & 637 & & \\ \downarrow & 2584 & 754 & 853 \\ -6^3 & 216 & & \\ \hline & 424 & 754 & \\ -3 \times 6^2 \times 3 & 324 & & \\ \hline & 100 & 754 \\ -3 \times 6 \times 3^2 & 162 & & \\ \hline & & 8454 \\ -3^3 & & 27 & \\ \hline & & 8427 & 853 \\ -3 \times 63^2 \times 7 & 83349 & \\ \hline & & & 9295 \\ -3 \times 63 \times 7^2 & & & 9261 \\ \hline & & & 343 \\ -7^3 & & & 343 \\ \hline & & & 0 \end{array} \]
Therefore, the cube root of 2584754853 is 637.
In simple words: To find the cube root of 2584754853, we use the Vedic method by breaking it into parts. We determine each digit of the root (6, 3, and 7) by applying specific Vedic formulas and subtracting from the remaining number, step by step.

🎯 Exam Tip: For longer numbers and multi-digit cube roots, ensure each intermediate term (\( 3a^2b \), \( 3ab^2 \), \( b^3 \)) is correctly subtracted from the evolving dividend. Practice helps with this complex multi-stage process.

Question 18. Find the cube root of 22665187.
Answer: We will find the cube root of 22665187 using the Vedic method. We group the digits into blocks from the right: 22 | 665 | 187. The largest perfect cube less than or equal to 22 is 8 (which is \( 2^3 \)). So, the first digit of the cube root is 2. The last digit of 22665187 is 7, so the last digit of its cube root must be 3. This implies the cube root is 283.
The calculation steps are as follows, using the components of 283:
\[ \begin{array}{r|rrr} & 283 & & \\ \downarrow & 22 & 665 & 187 \\ -2^3 & 8 & & \\ \hline & 14 & 665 & \\ -3 \times 2^2 \times 8 & 96 & & \\ \hline & 50 & 665 \\ -3 \times 2 \times 8^2 & 384 & & \\ \hline & 1225 & 187 \\ -8^3 & 512 & \\ \hline & 713 & 187 \\ -3 \times 28^2 \times 3 & 7056 & \\ \hline & & 758 \\ -3 \times 28 \times 3^2 & & 756 \\ \hline & & 27 \\ -3^3 & & 27 \\ \hline & & 0 \end{array} \]
Therefore, the cube root of 22665187 is 283.
In simple words: To find the cube root of 22665187 using the Vedic method, we divide the number into sections. We then find the cube root of the first part and use the last digit to find the final digit. Through a series of subtractions for the intermediate digits, we get 283 as the answer.

🎯 Exam Tip: When working with multi-digit cube roots, treat the first two digits of the root (e.g., '28') as a single unit when calculating terms for the third digit, adapting the formulas for accuracy.

Question 19. Find the cube root of 8615125.
Answer: We will find the cube root of 8615125 using the Vedic method. We group the digits into blocks of three from the right: 8 | 615 | 125. The largest perfect cube less than or equal to 8 is 8 (which is \( 2^3 \)). So, the first digit of the cube root is 2. The last digit of 8615125 is 5, so the last digit of its cube root must be 5. This implies the cube root is 205.
The calculation steps are as follows:
\[ \begin{array}{r|rrr} & 205 & & \\ \downarrow & 8 & 615 & 125 \\ -2^3 & 8 & & \\ \hline & 0 & 615 & \\ -3 \times 2^2 \times 0 & 0 & & \\ \hline & 615 & 125 \\ -3 \times 2 \times 0^2 & 0 & & \\ \hline & 615 & 125 \\ -0^3 & 0 & & \\ \hline & 615 & 125 \\ -3 \times 20^2 \times 5 & 6000 & \\ \hline & & 15125 \\ -3 \times 20 \times 5^2 & & 1500 \\ \hline & & 125 \\ -5^3 & & 125 \\ \hline & & 0 \end{array} \]
Therefore, the cube root of 8615125 is 205.
In simple words: To find the cube root of 8615125, we use the Vedic method by splitting the number into groups of three. We determine each digit (2, 0, and 5) by performing subtractions based on specific Vedic formulas at each step.

🎯 Exam Tip: Remember that a zero in the cube root (like in 205) means certain intermediate terms in the Vedic method will be zero, simplifying those particular subtractions.

Question 20. Find the cube root of 660776311.
Answer: We will find the cube root of 660776311 using the Vedic method. We group the digits into blocks of three from the right: 660 | 776 | 311. The largest perfect cube less than or equal to 660 is 512 (which is \( 8^3 \)). So, the first digit of the cube root is 8. The last digit of 660776311 is 1, so the last digit of its cube root must be 1. This implies the cube root is 871.
The calculation steps are as follows, using the components of 871:
\[ \begin{array}{r|rrr} & 871 & & \\ \downarrow & 660 & 776 & 311 \\ -8^3 & 512 & & \\ \hline & 148 & 776 & \\ -3 \times 8^2 \times 7 & 1344 & & \\ \hline & 143 & 76 & \\ -3 \times 8 \times 7^2 & 1176 & & \\ \hline & 2616 & 311 \\ -7^3 & 343 & \\ \hline & 2273 & 311 \\ -3 \times 87^2 \times 1 & 22707 & \\ \hline & & 261 \\ -3 \times 87 \times 1^2 & & 261 \\ \hline & & 1 \\ -1^3 & & 1 \\ \hline & & 0 \end{array} \]
Therefore, the cube root of 660776311 is 871.
In simple words: To find the cube root of 660776311, we use the Vedic method. We group the number into three-digit sections. We then find each digit of the root (8, 7, and 1) by applying specific Vedic formulas and subtracting them from the remaining numbers in each step.

🎯 Exam Tip: For complex cube root problems with multiple digits, correctly identifying the 'a' and 'b' values for each stage of subtraction (e.g., 'a' as 87 for the last digit '1') is crucial for accuracy.