Problem 18 - Algebra 2

Question

Write each expression in exponential form. $$\sqrt{-10}$$

Step-by-Step Solution

Answer

The exponential form of \(\sqrt{-10}\) is \(i * 10^{1/2}\).

Step 1: Identify the Negative Sign Under the Square Root

The square root of a negative number involves the imaginary unit 'i'. In general, for any positive real number 'a', \(\sqrt{-a}\) can be written as \(i\sqrt{a}\). Here, 'a' is 10.

Step 2: Write the Expression in Exponential Form

So \(\sqrt{-10}\) can be written as \(i\sqrt{10}\). Now, \(\sqrt{10}\) can be written as \(10^{1/2}\). Thus, \(\sqrt{-10}\) in exponential form becomes \(i * 10^{1/2}\).

Key Concepts

Understanding the Exponential Form

The exponential form is a powerful way to express numbers, especially when dealing with complex numbers and roots. In mathematics, exponential form refers to writing numbers using exponents. For example, a number like 10 can be expressed as an exponent by writing it as \(10^1\). Similarly, roots can also be expressed using fractional exponents. The square root of a number 'a' is written as \(a^{1/2}\). This is because the exponent \(1/2\) represents the square root operation.

When dealing with negative numbers under a square root, the expression usually involves an imaginary part. Imaginary numbers use the imaginary unit 'i', defined as the square root of -1. Therefore, if you have an expression like \(\sqrt{-10}\), it involves both the real and imaginary components, where the real part is written in exponential form as \(10^{1/2}\). This translates into combining both into an expression like \(i \cdot 10^{1/2}\).

• Exponential form helps in simplifying expressions, especially complex numbers.
• For any number 'a', \(\sqrt{a}\) can be expressed exponentially as \(a^{1/2}\).

Decoding the Square Root

The square root of a number is one of two equal factors of that number. For instance, the square root of 9 is 3, since \(3 \times 3 = 9\). In mathematical notation, the square root is denoted by the radical symbol \(\sqrt{}\). When dealing with positive numbers, things are straightforward. However, when a negative number is under the square root, it introduces the concept of imaginary numbers.

In the given problem, the square root of -10 is involved. Normally, the square root of a negative number isn't defined in the set of real numbers. Hence, we use the imaginary unit 'i', which is \(\sqrt{-1}\). Thus, the square root of -10 becomes \(i\sqrt{10}\). Such manipulation lets us handle the roots of negative numbers using "standard" mathematical approaches, by converting them into product forms containing imaginary numbers.

• Square roots determine the principal value, and for negative numbers, involve imaginary units.
• Imaginary numbers help extend the concept of square roots beyond positive values.

Exploring Complex Numbers

Complex numbers are a natural extension of real numbers, and they include both a real part and an imaginary part. They are usually expressed in the form \(a + bi\), where 'a' represents the real part, and 'b' is the imaginary coefficient of the imaginary unit 'i'.

In the context of our exercise, \(\sqrt{-10}\) converts naturally into a complex number expression \(i\cdot 10^{1/2}\). Here, the number doesn't have a real part (specifically due to the nature of roots with negatives under them), only an imaginary part stemming from the presence of 'i'.

• Complex numbers extend regular arithmetic to accommodate imaginary numbers.
• They are crucial for calculations in fields like engineering and physics.

Having a strong grasp of complex numbers opens up many avenues in advanced mathematics and helps solidify the understanding of concepts like oscillatory systems and wave mechanics.