Simplify square root of -1280-4*120*1500

Solution:

Step 1:

Calculate the product of $-4$, $120$, and $1500$.

Step 1.1:

First, multiply $-4$ by $120$ to get $-480$.$\sqrt{-1280 - 4 \cdot 120 \cdot 1500}$

Step 1.2:

Next, multiply $-480$ by $1500$ to obtain $-720000$.$\sqrt{-1280 - 720000}$

Step 2:

Combine $-1280$ and $-720000$ by subtraction.$\sqrt{-721280}$

Step 3:

Express $-721280$ as the product of $-1$ and $721280$.$\sqrt{-1 \cdot 721280}$

Step 4:

Separate the square root of the negative number into $\sqrt{-1}$ and $\sqrt{721280}$.$\sqrt{-1} \cdot \sqrt{721280}$

Step 5:

Replace $\sqrt{-1}$ with the imaginary unit $i$.$i \cdot \sqrt{721280}$

Step 6:

Decompose $721280$ into prime factors to simplify the square root.

Step 6.1:

Factor out $3136$ from $721280$.$i \cdot \sqrt{3136 \cdot 230}$

Step 6.2:

Recognize that $3136$ is a perfect square, $56^2$.$i \cdot \sqrt{(56)^2 \cdot 230}$

Step 7:

Extract the square root of the perfect square, $56$, from the radical.$i \cdot 56 \sqrt{230}$

Step 8:

Rearrange to place the coefficient $56$ in front of the imaginary unit $i$.$56i \sqrt{230}$

Knowledge Notes:

1. Complex Numbers: When dealing with square roots of negative numbers, we enter the realm of complex numbers. The fundamental unit of imaginary numbers is $i$, where $i^2 = -1$. This allows us to handle square roots of negative numbers as multiples of $i$.

2. Square Roots: The square root of a number $x$ is a number $y$ such that $y^2 = x$. When $x$ is negative, the square root is not a real number but an imaginary one.

3. Simplifying Square Roots: To simplify the square root of a number, we look for factors that are perfect squares, which can be taken out of the square root to simplify the expression.

4. Arithmetic Operations: The problem involves basic arithmetic operations such as multiplication and addition (in this case, represented as subtraction because of negative numbers).

5. Factorization: The process of breaking down a number into its constituent factors can greatly simplify the process of taking square roots, especially when those factors include perfect squares.

6. Order of Operations: When simplifying expressions, it's important to follow the correct order of operations. In this problem, we first perform multiplication and then addition (or subtraction), followed by simplifying the square root.

7. Imaginary Unit: The imaginary unit $i$ is used to represent the square root of $-1$. It's a fundamental concept in complex number theory and is essential for expressing square roots of negative numbers.

8. Algebraic Manipulation: The problem requires algebraic manipulation, including factoring and rearranging terms, to arrive at the simplest form of the expression.