Evaluate square root of -3^2+-4^2

Solution:

Step 1:

Consolidate the $+$ and $-$ signs into a single $-$ sign. When a plus sign is immediately followed by a minus sign, it is equivalent to a minus sign alone, since $1\cdot -1=-1$. Thus, we have $\sqrt{-3^2-4^2}$.

Step 2:

Begin simplifying the given expression.

Step 2.1:

Calculate the square of $3$, which is $\sqrt{-1\cdot 9-4^2}$.

Step 2.2:

Perform the multiplication of $-1$ with $9$, resulting in $\sqrt{-9-4^2}$.

Step 2.3:

Calculate the square of $4$, which is $\sqrt{-9-1\cdot 16}$.

Step 2.4:

Perform the multiplication of $-1$ with $16$, resulting in $\sqrt{-9-16}$.

Step 2.5:

Combine $-9$ and $-16$ to get $\sqrt{-25}$.

Step 3:

Express $-25$ as the product of $-1$ and $25$, which is $\sqrt{-1(25)}$.

Step 4:

Separate the square root of the product into the product of square roots, $\sqrt{-1}\cdot \sqrt{25}$.

Step 5:

Replace the square root of $-1$ with the imaginary unit $i$, giving us $i\cdot \sqrt{25}$.

Step 6:

Express $25$ as the square of $5$, which is $i\cdot \sqrt{5^2}$.

Step 7:

Extract terms from under the square root, assuming they represent positive real numbers, to get $i\cdot 5$.

Step 8:

Rearrange the terms to place the numerical coefficient before the imaginary unit, resulting in $5i$.

Knowledge Notes:

The problem involves evaluating the square root of a sum of squares with negative signs. This is a common problem in complex number arithmetic. Here are the relevant knowledge points:

1. Combining Signs: The combination of a plus sign followed by a minus sign can be simplified to a single minus sign.

2. Exponentiation: Squaring a number means multiplying the number by itself.

3. Multiplication by -1: Multiplying a number by -1 changes its sign.

4. Square Root of a Negative Number: The square root of a negative number involves the imaginary unit $i$, where $i^2=-1$.

5. Separation of Square Roots: The square root of a product can be expressed as the product of the square roots of the individual factors.

6. Extraction of Square Roots: When extracting the square root of a perfect square, the result is the positive number that, when squared, gives the original number.

7. Imaginary Numbers: Imaginary numbers are expressed as a real number multiplied by the imaginary unit $i$.

8. Order of Terms: In complex numbers, it is conventional to write the coefficient before the imaginary unit (e.g., $5i$ instead of $i5$).

In this problem, we use these concepts to simplify the expression step by step, ultimately arriving at a complex number in standard form.