Square Root of 1.75

The square root is the inverse operation of squaring a number. Since 1.75 is not a perfect square, its square root is expressed in both radical and exponential forms. In radical form, it is expressed as √1.75, whereas in exponential form, it is (1.75)^(1/2). The square root of 1.75 is approximately 1.3228756555323, which is an irrational number because it cannot be expressed as a fraction where both numerator and denominator are integers, and the denominator is not zero.

The prime factorization method is typically used for perfect squares. However, for non-perfect squares like 1.75, we use the long division method and approximation method. Let us explore these methods:

• Long division method
• Approximation method

The long division method is effective for finding the square roots of non-perfect squares. Here is how to find the square root of 1.75 using this method:

Step 1: Start by placing a decimal point in the dividend, 1.75, and pair the digits from the decimal point.

Step 2: Consider the number 1. The largest square less than or equal to 1 is 1 (1^2=1). Subtract 1 from 1 to get a remainder of 0.

Step 3: Bring down 75 to make it 175.

Step 4: Double the previous divisor (1) to get the new divisor, 2.

Step 5: Find a number, n, such that 2n * n is less than or equal to 175. Here, n is 6 because 26 * 6 = 156.

Step 6: Subtract 156 from 175 to get 19, and bring down double zeros to make it 1900.

Step 7: Double the previous result (26) to get 52. Find a number, n, such that 52n * n is less than or equal to 1900. Here, n is 3 because 523 * 3 = 1569.

Step 8: Continue this process to refine the value to desired decimal places.

The approximate square root of 1.75 is 1.3228756555323.

The approximation method provides an easy way to estimate square roots:

Step 1: Identify the closest perfect squares around 1.75. These are 1 (1^2) and 4 (2^2). Therefore, √1.75 is between √1 (1) and √4 (2).

Step 2: Use linear interpolation to estimate. Calculate (1.75 - 1) / (4 - 1) = 0.25.

Step 3: Add this result to the lower limit, 1. So, 1 + 0.25 = 1.25, a rough estimate. Further refinement yields 1.3228756555323.

Students often make various mistakes when finding square roots. Here are common mistakes and tips to avoid them:

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: the square root of 9 is 3, since 3 × 3 = 9.
   
• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction; its decimal goes on forever without repeating. Example: π (pi) is an irrational number.
   
• Approximation: An approximation is a value or number close to the actual value but not exact, often used when exact values are difficult to ascertain.
   
• Exponential form: In mathematics, exponential form refers to numbers written with a base and an exponent. For instance, 2^3 is the exponential form of 8.
   
• Linear interpolation: A method of estimating an unknown value by using two known values on either side of the unknown value and assuming a linear relationship between them.