Square Root of 3/2

The square root is the inverse operation of squaring a number. The number 3/2 is not a perfect square. The square root of 3/2 is expressed in both radical and exponential form. In radical form, it is expressed as √(3/2), whereas (3/2)^(1/2) in exponential form. √(3/2) is an irrational number because it cannot be expressed as a simple fraction of two integers.

For non-perfect square numbers, the prime factorization method is not applicable. Instead, methods like the long-division method and approximation method are used. Let us explore these methods:

• Long division method
• Approximation method

The long division method is useful for finding the square roots of non-perfect square numbers. Here is how we can calculate the square root of 3/2 using this method:

Step 1: Convert the fraction 3/2 to a decimal, which is 1.5.

Step 2: Use the long division method to find the square root of 1.5.

Step 3: Group the digits of 1.5, considering it as 150 in the long division format.

Step 4: Find the largest number whose square is less than or equal to 150.

Step 5: Continue the long division process until you reach the desired level of precision.

The square root of 1.5 is approximately 1.2247, so √(3/2) ≈ 1.2247.

The approximation method is another approach to finding square roots. This method is relatively easy for estimating the square root of a number. Here is how we can find the square root of 3/2 using approximation:

Step 1: Convert the fraction 3/2 to a decimal, which is 1.5.

Step 2: Identify two perfect squares between which 1.5 falls. The perfect squares are 1 (1^2) and 4 (2^2), so √1.5 is between 1 and 2.

Step 3: Use interpolation to estimate √1.5. The difference between 1.5 and 1 is 0.5, and the difference between 4 and 1 is 3.

Step 4: Calculate the approximate value using interpolation: (1.5 - 1) / (4 - 1) = 0.5 / 3 = 0.1667.

Step 5: Add this value to the lower bound of 1 to estimate √1.5 ≈ 1.1667. For more precision, further calculations can be done.

The square root of 3/2 is approximately 1.2247.

• 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 4 is 2, because 2 × 2 = 4.

• Irrational number: An irrational number cannot be expressed as a simple fraction; its decimal form is non-repeating and non-terminating. Example: √2 is irrational.

• Rational number: A rational number can be expressed as a fraction with an integer numerator and a non-zero integer denominator. Example: 3/2 is rational.

• Decimal: A decimal represents a non-integer number using place value and a decimal point. Example: 1.5 is a decimal representation of 3/2.

• Interpolation: Interpolation is a method of estimating unknown values that fall between known values. It is used in approximating square roots, like estimating √1.5.