Problem 111 - Elementary and Intermediate Algebra

Question

Simplify each radical expression, if possible. Assume all variables are unrestricted. $$ \sqrt{400 m^{16} n^{2}} $$

Step-by-Step Solution

Answer

The simplified form is \( 20m^8n \).

Step 1: Simplify the Square Root of the Constant

First, identify and simplify the square root of the constant 400. Since 400 is a perfect square (as it is equal to 20 squared), we have \( \sqrt{400} = 20 \).

Step 2: Simplify the Square Root of the Variable \(m^{16}\)

Recognize that \(m^{16}\) can be expressed as \((m^8)^2\). Hence, the square root of \(m^{16}\) is \(m^8\). This is because \( \sqrt{m^{16}} = m^{16/2} = m^8 \).

Step 3: Simplify the Square Root of the Variable \(n^2\)

Since \(n^2\) is a perfect square, its square root is simply \(n\). Thus, \( \sqrt{n^2} = n \).

Step 4: Combine the Results

Now, combine the simplified results from the previous steps. The simplified expression is \(20 \times m^8 \times n\). So, the simplified form of \( \sqrt{400 m^{16} n^{2}} \) is \(20m^8n\).

Key Concepts

Perfect Square

A perfect square is a number that can be expressed as the square of an integer. This concept is crucial when working with radicals because it allows for simplification. When a number is a perfect square, its square root is an integer. In our exercise, the number 400 is a perfect square because it equals \(20^2\). Thus, \(\sqrt{400} = 20\).
Recognizing perfect squares can make your calculations easier and faster. The trick is to remember some common perfect squares such as:

• \(1 = 1^2\)
• \(4 = 2^2\)
• \(9 = 3^2\)
• \(16 = 4^2\)
• \(25 = 5^2\)
• \(100 = 10^2\)

Identifying these numbers quickly enables you to simplify radical expressions much more efficiently.

Variables

Variables can be trickier than constants when simplifying radicals, but similar principles apply. Variables with even exponents are usually perfect squares, which simplifies the process. For example, the term \(m^{16}\) in the expression is a perfect square because \(m^{16} = (m^8)^2\). This means \(\sqrt{m^{16}} = m^8\).
Similarly, another variable in this expression, \(n^2\), is also a perfect square because its exponent is 2, which gives \(\sqrt{n^2} = n\).
When simplifying variables:

• Take note if the exponent is even, the variable is likely a perfect square.
• Divide the exponent by 2 to find the new exponent after taking the square root.
• If the exponent is odd, the expression is not a perfect square and cannot be simplified directly in this manner.

Simplification of Radicals

Simplifying radicals involves finding equivalent expressions where the radical sign is eliminated as much as possible. This process often includes:

• Identifying perfect squares within the radicand.
• Simplifying both constant and variable components separately.

In the given expression \(\sqrt{400m^{16}n^{2}}\), simplifying involves breaking down to the most basic form using perfect square principles and variable exponents.
Start with the constant, as seen with \(\sqrt{400} = 20\). Then handle each variable separately. For \(m^{16}\), recognize the even exponent and apply the square root to get \(m^{8}\). Similarly, \(\sqrt{n^2} = n\).
Finally, you combine these results into one expression: \(20 \times m^8 \times n\). The goal of simplification is to reduce complexity and make the expression easier to compute, while retaining equality with the original form.