Square Root Problems (General Method)

These problems can be solved using any suitable method, such as prime factorization or estimation:

• Find $\sqrt{2025}$

• Find $\sqrt{4096}$

• Find $\sqrt{14641}$

• Find $\sqrt{160000}$

• Find $\sqrt{531441}$

Square Root Problems (Long Division Method)

The following problems specifically require the use of the long division method:

• Find $\sqrt{3136}$

• Find $\sqrt{10816}$

• Find $\sqrt{20736}$

• Find $\sqrt{234256}$

• Find $\sqrt{998001}$

Group 1: Using Flexible Methods

For these problems, you can use Prime Factorization (breaking the number into prime factors) or Estimation.

• Find $\sqrt{2025}$

  • Explanation: Since the number ends in 25, we know the square root must end in 5. By prime factorization, $2025 = 3^4 \times 5^2$. Taking the square root gives $3^2 \times 5 = 9 \times 5 = 45$.

• Find $\sqrt{4096}$

  • Explanation: This is a power of 2 ($2^{12}$). The square root is $2^6$, which equals 64.

• Find $\sqrt{14641}$

  • Explanation: This number follows the pattern of powers of 11. Specifically, $11^4 = 14641$, so the square root is $11^2 = 121$.

• Find $\sqrt{160000}$

  • Explanation: For numbers with an even number of trailing zeros, you can take the square root of the non-zero part and halve the number of zeros. $\sqrt{16} = 4$ and four zeros become two zeros, resulting in 400.

• Find $\sqrt{531441}$

  • Explanation: This large number can be solved by grouping digits from right to left (41, 14, 53). Since it ends in 1, the root ends in 1 or 9. Through factorization or estimation, we find it is $729^2$.

Group 2: Using the Long Division Method

The long division method is best for large numbers where prime factors aren’t immediately obvious. It involves grouping digits in pairs from the decimal point.

• Find $\sqrt{3136}$

  • Explanation: Grouped as 31 and 36. The largest square less than 31 is 25 ($5^2$). After subtracting and bringing down 36, we find the next digit is 6. The answer is 56.

• Find $\sqrt{10816}$

  • Explanation: Grouped as 1, 08, and 16. We start with 1 ($1^2=1$). Following the long division steps for the subsequent groups 08 and 16, we arrive at 104.

• Find $\sqrt{20736}$

  • Explanation: Grouped as 2, 07, and 36. Starting with 1 ($1^2=1$), then working through the remainder and the next groups, the calculation yields 144.

• Find $\sqrt{234256}$

  • Explanation: Grouped as 23, 42, and 56. The nearest square to 23 is 16 ($4^2$). Through the iterative division process, the result is 484.

• Find $\sqrt{998001}$

  • Explanation: Grouped as 99, 80, and 01. The largest square less than 99 is 81 ($9^2$). Continuing the long division steps, the square root is found to be 999.