Problem 27 - Algebra 2

Question

Simplify. $$ \sqrt{-81} $$

Step-by-Step Solution

Answer

\( \sqrt{-81} = 9i \).

Step 1: Identify the Negative Root

The expression \( \sqrt{-81} \) involves taking the square root of a negative number, which is not possible in the set of real numbers. We need to rewrite the expression in terms of imaginary numbers. Recall that \( i \) is defined as \( \sqrt{-1} \).

Step 2: Rewrite the Expression

Rewrite \( \sqrt{-81} \) as \( \sqrt{-1 \times 81} \). According to the properties of square roots, this is equivalent to \( \sqrt{-1} \times \sqrt{81} \).

Step 3: Simplify with Imaginary Number

Substitute \( i \) for \( \sqrt{-1} \), giving us \( i \times \sqrt{81} \).

Step 4: Calculate the Square Root

Determine the value of \( \sqrt{81} \). Since \( 81 = 9^2 \), we have \( \sqrt{81} = 9 \).

Step 5: Combine the Factors

Combine the expressions to get \( i \times 9 \), which simplifies to \( 9i \). Therefore, \( \sqrt{-81} = 9i \).

Key Concepts

Square Roots

Square roots are mathematical expressions that allow us to find a number which, when multiplied by itself, gives the original number. For example, the square root of 81 is 9 because 9 times 9 equals 81. The symbol for square roots is \( \sqrt{\cdot} \). When dealing with positive numbers, finding square roots is straightforward. However, the challenge arises when we encounter negative numbers under the square root.
- Square roots only work naturally with non-negative numbers in the set of real numbers.- The concept extends to negative numbers using imaginary numbers, specifically introducing the idea of \( i \), which is defined as \( \sqrt{-1} \).
Understanding square roots is fundamental because it is a stepping stone toward more complex mathematical concepts, such as dealing with negative numbers and complex numbers.

Negative Numbers

Negative numbers are just as important in mathematics as positive numbers. They are numbers less than zero and are found on the left side of a number line. Negative numbers often represent values such as debts, temperatures below zero, or anything that indicates a reduction.
- When it comes to square roots, negative numbers present a unique challenge because there is no real number whose square is negative.- This is where the concept of imaginary numbers comes into play, allowing us to take the square root of a negative number by using \( i \).
The notion of negative numbers is essential for understanding how imaginary numbers function, which is our next topic.

Complex Numbers

Complex numbers are an extension of the real numbers and include both a real part and an imaginary part. They are written in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit defined as \( \sqrt{-1} \).
- Imaginary numbers arise from the need to take square roots of negative numbers.- In the expression \( \sqrt{-81} \), we rewrite it using the imaginary unit \( i \, \) as \( \sqrt{-1 \times 81} = i \times \sqrt{81} \).- This allows us to simplify \( \sqrt{-81} \) to \( 9i \), making it a complex number since it involves the imaginary unit.
Complex numbers are fundamental in advanced mathematics, engineering, and physics, allowing computations that aren't possible using only real numbers. Understanding them helps broaden the scope of mathematical problem-solving to include solutions like the one addressed in the original exercise.