Square Root of -147

The square root is the inverse of the square of the number. Since -147 is a negative number, it does not have a real square root because a real number squared is always non-negative. Instead, the square root of -147 is expressed in terms of an imaginary number. In radical form, it is expressed as √-147, which can be written as i√147 in terms of real and imaginary components, where i is the imaginary unit (i² = -1). The approximate value of √147 is 12.124, so the square root of -147 is approximately 12.124i.

To find the square root of a negative number, we use imaginary numbers. Here, we will illustrate how to express the square root of -147 using imaginary numbers:

Step 1: Express the negative number in terms of a positive number and the imaginary unit: √-147 = √(147) × √(-1) = √147 × i

Step 2: Calculate the square root of 147, which is approximately 12.124.

Step 3: Combine the result with the imaginary unit: √-147 = 12.124i

To find the square root of 147, we can use the prime factorization method for the positive part of the number:

Step 1: The prime factorization of 147 is 3 × 7 × 7 or 3 × 7².

Step 2: Pair the prime factors: Since we have a pair of 7s, we can take one 7 out of the square root.

Step 3: The square root of 147 in simplest form is expressed as √147 = √(3 × 7²) = 7√3.

Thus, the square root of -147 is 7√3i.

The long division method is typically used for finding square roots of non-perfect squares. For -147, we are interested in the imaginary square root:

Step 1: Consider the positive part, 147, and find its square root using long division, which is approximately 12.124.

Step 2: Since -147 is negative, the square root is expressed as an imaginary number: √-147 = 12.124i.

To approximate the square root of -147, we use the square root of 147 and express it in terms of an imaginary number:

Step 1: Approximate √147 using the closest perfect squares, 144 and 169. √147 lies between 12 and 13.

Step 2: Calculate the decimal approximation: (147 - 144) / (169 - 144) = 3 / 25 = 0.12

Step 3: Add the decimal approximation to the lower bound: 12 + 0.12 = 12.12

Step 4: Therefore, the square root of -147 is approximately 12.12i.

• Imaginary Unit: Represented by i, it is the square root of -1, used to express square roots of negative numbers.

• Square Root: The value that, when multiplied by itself, gives the original number. For negative numbers, it includes the imaginary unit.

• Prime Factorization: The expression of a number as the product of its prime factors, used for simplifying square roots.

• Approximation: A method of finding a near value, applied to non-perfect square roots for estimating decimal values.

• Real Number: A value that represents a quantity along a continuous line, excluding imaginary numbers like those involving i.