write the expression in the form bi, where b is the real number squareroot-16 squareroot-25...

Question:

Write the expression in the form bi, where b is the real number. {eq}\sqrt{-16} \sqrt{-25} \sqrt{-3} \sqrt{-5} {/eq}

Square Root of a Negative Number

In order to simplify the square root of a negative number, we need to apply a specific definition: {eq}\sqrt{-1} = i {/eq}. This allows us to write the square root of a negative number in the form {eq}bi {/eq}, where b is a real number.

Answer and Explanation:

We can write each of these square roots as an imaginary number. To do so, we need to apply the definition {eq}\sqrt{-1} = i {/eq} and simplify the rest of the square root. Let's go through this procedure for each of the four square roots in this problem.

First, the square root of negative 16. We can simplify this as follows.

{eq}\begin{align*} \sqrt{-16} &= \sqrt{-1 \cdot 16} \\ &= \sqrt{-1} \cdot \sqrt{16}\\ &= i \sqrt {16}\\ &= 4i \end{align*} {/eq}


We can simplify the next three square roots in the same way, by bringing out the imaginary constant and evaluating the rest of the square root, if possible.

{eq}\begin{align*} \sqrt{-25} &= \sqrt{-1\cdot 25}\\ &= \sqrt{-1} \cdot \sqrt{25}\\ &= i \sqrt{25}\\ &= 5i \end{align*} {/eq}


The next two square roots will still contain a square root in them, as we cannot simplify the square root of 3 or the square root of 5. To avoid confusion, we usually write the imaginary constant in front of the square root in this case so that we don't accidentally put it underneath.

{eq}\begin{align*} \sqrt{-3} &= \sqrt{-1 \cdot 3}\\ &= \sqrt{-1} \cdot \sqrt 3\\ &= i \sqrt 3\\ \sqrt{-5} &= \sqrt{-1 \cdot 5}\\ &= \sqrt{-1} \cdot \sqrt 5\\ &= i \sqrt 5\\ \end{align*} {/eq}


Since our overall expression is the product of these four square roots, we can now multiply this entire expression together to yield our solution. Since {eq}i^2 = -1 {/eq}, we can extend this to draw the conclusion that {eq}i^4 = 1 {/eq}, which will help us reach our solution.

{eq}\begin{align*} \sqrt{-16} \sqrt{-25} \sqrt{-3} \sqrt{-5} &= 4i \cdot 5i \cdot i \sqrt 3 \cdot i \sqrt 5\\ &= i^4 ( 4 \cdot 5 \cdot \sqrt 3 \cdot \sqrt 5)\\ &= 1 (20 \sqrt{15})\\ &= 20 \sqrt {15} \end{align*} {/eq}


Learn more about this topic:

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What is an Imaginary Number?

from Math 101: College Algebra

Chapter 5 / Lesson 1
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