Find the Square Root of 90 by Estimation Method

1. Identify the Closest Perfect Squares

   The number 90 lies between the perfect squares 81 (\(9^2\)) and 100 (\(10^2\)). Hence, the square root of 90 is between 9 and 10.

2. Initial Approximation

   Since 90 is closer to 81 than 100, start with an initial estimate slightly above 9. Let’s use 9.5 as the initial estimate.

3. Refine the Estimate

   To refine the estimate, use the average method:

   • Divide 90 by the initial estimate: \( \frac{90}{9.5} \approx 9.474 \)
   • Find the average of the initial estimate and the quotient: \( \frac{9.5 + 9.474}{2} \approx 9.487 \)

   This gives a refined estimate of approximately 9.487.

4. Iterative Refinement

   Repeat the process if a more accurate value is needed. For most practical purposes, 9.487 is sufficiently precise.

Newton-Raphson Method

Another approach to find a more precise value is the Newton-Raphson method, which iteratively improves the estimate:

1. Choose an initial estimate (\(x_0\)). Let’s use 9.
2. Apply the formula: \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \)
3. For \( f(x) = x^2 - 90 \), the derivative \( f'(x) = 2x \).
4. Using \( x_0 = 9 \):
   \( f(9) = 9^2 - 90 = -9 \)
   \( f'(9) = 2 \times 9 = 18 \)
   \( x_1 = 9 - \frac{-9}{18} = 9.5 \)
5. Repeat with \( x_1 = 9.5 \) to get an even closer approximation.

By using these estimation methods, you can find that the square root of 90 is approximately 9.487, which can be verified through various methods like the average method or Newton-Raphson method.

The square root of 90 is an irrational number, meaning it cannot be expressed exactly as a simple fraction and its decimal representation is non-repeating and non-terminating.

Use these methods to estimate square roots of non-perfect squares effectively in your calculations.

Newton-Raphson Method

Another approach to find a more precise value is the Newton-Raphson method, which iteratively improves the estimate:

1. Choose an initial estimate (\(x_0\)). Let’s use 9.
2. Apply the formula: \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \)
3. For \( f(x) = x^2 - 90 \), the derivative \( f'(x) = 2x \).
4. Using \( x_0 = 9 \):
   \( f(9) = 9^2 - 90 = -9 \)
   \( f'(9) = 2 \times 9 = 18 \)
   \( x_1 = 9 - \frac{-9}{18} = 9.5 \)
5. Repeat with \( x_1 = 9.5 \) to get an even closer approximation.

By using these estimation methods, you can find that the square root of 90 is approximately 9.487, which can be verified through various methods like the average method or Newton-Raphson method.

The square root of 90 is an irrational number, meaning it cannot be expressed exactly as a simple fraction and its decimal representation is non-repeating and non-terminating.

Use these methods to estimate square roots of non-perfect squares effectively in your calculations.

By using these estimation methods, you can find that the square root of 90 is approximately 9.487, which can be verified through various methods like the average method or Newton-Raphson method.

The square root of 90 is an irrational number, meaning it cannot be expressed exactly as a simple fraction and its decimal representation is non-repeating and non-terminating.

Use these methods to estimate square roots of non-perfect squares effectively in your calculations.

Estimating the square root of 90 using the estimation method is a valuable skill that simplifies complex calculations. This method involves finding the two perfect squares between which 90 lies and then interpolating to get a close approximation. For example, 90 is between 81 (9²) and 100 (10²), so its square root is between 9 and 10. This step-by-step guide will demonstrate the process in a detailed and easy-to-understand manner.

The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical terms, if \( a \) is the square root of \( b \), then \( a \times a = b \). For example, the square root of 16 is 4 because \( 4 \times 4 = 16 \). The square root is denoted by the radical symbol \( \sqrt{} \). Thus, the square root of 90 is represented as \( \sqrt{90} \).

Every positive number has two square roots: a positive and a negative root. For example, both 3 and -3 are square roots of 9 because \( 3 \times 3 = 9 \) and \( -3 \times -3 = 9 \). However, in most contexts, the term "square root" refers to the positive root.

Square roots are essential in various mathematical applications, including solving quadratic equations, understanding geometric properties, and in real-world scenarios such as physics and engineering. The square root function is fundamental in algebra and beyond.

To summarize, the square root of a number \( x \) is the number \( y \) such that \( y^2 = x \). It is a critical concept that underpins many mathematical theories and practical applications.

Estimating the square root of a number like 90 can be approached using several methods. Below are detailed steps for two commonly used estimation methods: the Decimal Method and the Closest Perfect Square Method.

1. Decimal Method

This method involves finding the nearest perfect square to the number and then estimating the decimal part of the square root.

1. Identify the nearest perfect squares to 90. These are 81 (9 × 9) and 100 (10 × 10).
2. Since 90 is closer to 81, start with the square root of 81, which is 9.
3. Calculate the decimal part: Estimate how much more than 9 the square root of 90 might be. Considering the difference, you can approximate the decimal part to be around 0.5.
4. Combine the whole number and decimal part to get an estimated square root: 9 + 0.5 = 9.5.

2. Closest Perfect Square Method

This method uses the two closest perfect squares to narrow down the estimation.

1. Identify the two closest perfect squares to 90, which are 81 (9 × 9) and 100 (10 × 10).
2. Determine that the square root of 90 lies between 9 and 10.
3. Use a midpoint estimate: Average 9 and 10 to get 9.5.
4. Refine the estimate: Knowing 90 is closer to 81 than to 100, adjust the estimate slightly downwards to approximately 9.4.

3. Averaging Method

This method involves averaging and refining the guess iteratively.

1. Start with an initial guess, such as 9.5.
2. Divide 90 by the initial guess: 90 ÷ 9.5 ≈ 9.47.
3. Average the initial guess and the result: (9.5 + 9.47) ÷ 2 ≈ 9.485.
4. Repeat the process until the estimate stabilizes: Continue dividing and averaging until the result is consistent. The final estimate will be around 9.487.

Conclusion

Estimating the square root of 90 can be done effectively using any of these methods. Each method provides a good approximation, with the final estimated value being close to 9.487. Understanding these techniques is useful for quick calculations when precision is not critical.

Estimating the square root of a number using perfect squares is an effective method. This method involves finding the two closest perfect squares that the given number lies between, then estimating the value based on their relationship. Here’s how you can estimate the square root of 90 using perfect squares:

1. Identify the nearest perfect squares: Find the perfect squares closest to 90. The nearest perfect squares are 81 (9²) and 100 (10²).
2. Determine the square roots: The square roots of 81 and 100 are 9 and 10, respectively.
3. Estimate the square root: Since 90 is closer to 81 than to 100, we estimate the square root of 90 to be slightly more than 9. To refine the estimate, consider the positions of 90 within the range:
   • The difference between 90 and 81 is 9.
   • The difference between 100 and 81 is 19.
   • The fraction of the distance from 81 to 90 compared to the total distance from 81 to 100 is 9/19, or approximately 0.47.
4. Calculate the estimate: Add this fraction to the lower square root value (9):

   \[ \sqrt{90} \approx 9 + 0.47 = 9.47 \]

Thus, using perfect squares, we estimate the square root of 90 to be approximately 9.47. This method provides a reasonable approximation without the need for a calculator.

The decimal method is a systematic approach to estimate the square root of a number by refining guesses to achieve a desired level of accuracy. Below are the steps to find the square root of 90 using this method:

1. Identify the Perfect Squares: Determine the two perfect squares between which 90 lies. Since \(81 = 9^2\) and \(100 = 10^2\), the square root of 90 will be between 9 and 10.
2. Initial Estimate: Start with an initial estimate. Since 90 is closer to 81 than to 100, an initial guess can be 9.5.
3. Refinement: Refine the estimate using the average method. Take the average of the initial guess and the quotient of 90 divided by the initial guess: \[ \text{New Estimate} = \frac{9.5 + \frac{90}{9.5}}{2} \] \[ \text{New Estimate} = \frac{9.5 + 9.4737}{2} \approx 9.4868 \]
4. Iteration: Repeat the refinement process with the new estimate until the desired accuracy is achieved. In this case, the process converges quickly, showing that the square root of 90 is approximately 9.4868.

This method provides a practical way to estimate square roots, leveraging basic arithmetic and averages to achieve accurate results.

The average method, also known as the iterative method, is a straightforward way to estimate the square root of a number. This method involves taking an initial guess and improving it through successive approximations.

To find the square root of 90 using the average method, follow these steps:

1. Make an initial guess. Let's choose 9 as our initial guess since we know that 9² = 81, which is close to 90.

2. Divide 90 by your guess.
   
   \( \frac{90}{9} = 10 \)

3. Find the average of your guess and the result from step 2.
   
   \(\frac{9 + 10}{2} = 9.5 \)

4. Use the result from step 3 as your new guess and repeat the process:
   
   \( \frac{90}{9.5} \approx 9.47368 \)
   
   \( \frac{9.5 + 9.47368}{2} \approx 9.48684 \)

5. Continue this process until the result stabilizes. After a few iterations, the value will converge to approximately 9.48683.

This iterative approach provides a reasonably accurate estimate of the square root of 90, demonstrating the utility and simplicity of the average method for non-perfect squares.

The Prime Factorization Method involves breaking down the number 90 into its prime factors and then simplifying under the square root sign. While this method is less practical for precise estimation, it provides a clear mathematical approach.

1. Start with the prime factorization of 90:

   \(90 = 2 \times 3^2 \times 5\)

2. Rewrite the square root of 90 using its prime factors:

   \(\sqrt{90} = \sqrt{2 \times 3^2 \times 5}\)

3. Separate the perfect square from the other factors:

   \(\sqrt{90} = \sqrt{3^2 \times 2 \times 5} = \sqrt{3^2} \times \sqrt{2 \times 5}\)

4. Simplify the square root of the perfect square:

   \(\sqrt{3^2} = 3\)

5. Combine the simplified term with the remaining factors under the square root:

   \(\sqrt{90} = 3 \times \sqrt{10}\)

6. Approximate the value of \(\sqrt{10}\):

   Since \(\sqrt{10} \approx 3.16\), we get:

   \(\sqrt{90} \approx 3 \times 3.16 = 9.48\)

Thus, by using the prime factorization method, we can estimate the square root of 90 to be approximately 9.48. This method illustrates the process of breaking down the number into manageable parts and simplifying them step by step.