Problem 68 - Algebra 2

Question

Simplify each expression. $$ \sqrt{180 n^{5}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\sqrt{180 n^{5}}\) is \(6n^2\sqrt{5n}\).

Step 1: Factorize the numerical part

180 is not a perfect square. So the first step is to factorize it as much as we can, so that we can take out the square root of any perfect squares. Now, \(180 = 2^2*3^2*5\). Hence, \(\sqrt{180} = \sqrt{2^2*3^2*5} = 2*3\sqrt{5} = 6\sqrt{5}\).

Step 2: Simplify the variable part

The variable part \(n^5\) is not a perfect square as well, but \(n^4\) is, since it's \(n^2\) to the power of 2 and \(n^1\) can remain under the radicand. Hence, we can express \(n^5\) as \((n^2)^2*n\), so \(\sqrt{n^5}= \sqrt{(n^2)^2*n} = n^2*\sqrt{n}\).

Step 3: Combine the simplified numerical and variable part

Combine the numerical and variable part that we simplified in step 1 and step 2: \( \sqrt{180 n^{5}} = 6\sqrt{5} * n^2*\sqrt{n} = 6n^2\sqrt{5n}\).

Key Concepts

Factoring

Factoring is a crucial technique in simplifying radical expressions, and it involves breaking down numbers or algebraic expressions into their basic components, known as factors. In the context of the given problem, you start by factorizing the number 180. This means rewriting 180 as a product of its prime factors. Prime factors are numbers that are only divisible by 1 and themselves, such as 2, 3, 5, and so forth.
For 180, you can break it down as follows:

• 180 is divisible by 2, giving you 90.
• 90 is divisible by 2 again, resulting in 45.
• 45 is divisible by 3, which gives you 15.
• Finally, 15 is divisible by 3, ending up with 5, which is prime itself.

Putting it all together, 180 can be expressed as \(2^2 \times 3^2 \times 5\).
This step allows us to identify perfect square factors within the expression, which is a significant step toward simplifying the expression under the square root. By preparing the expression in terms of its factors, you can easily bring out any factor that is a perfect square.

Square Roots

Understanding square roots is essential when it comes to simplifying radical expressions. A square root asks, "What number, when multiplied by itself, gives this number?"
For example, the square root of 4 is 2, because \(2 \times 2 = 4\). Similarly, if you have \(x^2\), \(\sqrt{x^2}\) equals \(x\), representing the original number before squaring.
When solving the example problem, you handle \(\sqrt{180}\). After factoring, you see \(180 = 2^2 \times 3^2 \times 5\). Knowing that \(\sqrt{2^2} = 2\) and \(\sqrt{3^2} = 3\), you utilize these facts:

• You can "take out" these perfect squares from under the square root: \(\sqrt{2^2 \times 3^2 \times 5}\).
• Doing so, you get \(2 \times 3\), resulting in 6.

So this gives you \(6 \sqrt{5}\). The factors that cannot be "taken out," like 5 here, remain inside the radical. This simplification is fundamental and is applied to both numbers and variable expressions, like \(n^5\) in the problem.

Perfect Squares

Perfect squares are numbers or expressions that have a "whole" square root, meaning they can be completely solved or simplified. A perfect square is the result of squaring a whole number or variable like \(a^2\), making \(a\) the square root.
In the problem example, during the breakdown of \(n^5\), you identify \((n^2)^2\) as a perfect square. Here's why:

• \(n^2\) times itself gives you \((n^2)^2\), which simplifies perfectly to \(n^2\).
• This allows you to "take out" \(n^2\) from the square root.

What's left under the square root is \(n\), since \(n^5\) was originally \((n^2)^2 \times n\). Understanding perfect squares helps you simplify expressions by identifying and using factors that fit this criterion. They enable you efficiently to simplify radical expressions, making them more straightforward and manageable by addressing both numerical and variable aspects of the equation.