Problem 41 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ 4 \sqrt{18 a^{5} b^{17}} $$

Step-by-Step Solution

Answer

Question: Simplify the radical expression 4√(18a^5b^17). Answer: $$ 4 \sqrt{18 a^{5} b^{17}} = 12a^2b^8 \sqrt{2ab} $$

Step 1: Simplify the coefficient inside the square root

To simplify the coefficient inside the square root, we will find the prime factors of 18 and consider only those factors that can be paired. The square root of paired factors can then be taken outside the root. $$ 4 \sqrt{18} = 4 \sqrt{2 \times 3^2} = 4 \times 3 \sqrt{2} = 12 \sqrt{2} $$

Step 2: Simplify the variable part inside the square root

Now, we will simplify the variables inside the square root. We will take out those a and b with even powers outside the square root. $$ \sqrt{a^5 b^{17}} = \sqrt{a^4 a b^{16} b} = a^2 b^8 \sqrt{ab} $$

Step 3: Combine constants and variables

Lastly, we will combine the constants and variables we have simplified in steps 1 and 2: $$ 12 \sqrt{2} \times a^2 b^8 \sqrt{ab} = 12a^2b^8 \cdot \sqrt{2ab} $$ Thus, the simplified expression is: $$ 4 \sqrt{18 a^{5} b^{17}} = 12a^2b^8 \sqrt{2ab} $$

Key Concepts

Prime Factorization

Prime factorization is an essential step in simplifying radical expressions. It involves breaking down a number into its basic building blocks, which are prime numbers. Think of it as turning a large Lego structure into its individual bricks. For example, with 18, you would divide it into prime factors: 2 and 3. More specifically, 18 can be expressed as \(2 \times 3^2\). By identifying these prime factors, we see that \(3^2\) can be "paired" and brought out of the square root as a single 3. This is important because it helps reduce the complexity of the expression inside the root.

Square Root

The square root tells us to find what number, when multiplied by itself, equals the given number inside the radical. In our example, we are dealing with the square root of numbers and variables, such as \( \sqrt{18} \).

• The square root of a perfect square like \(3^2\) is straightforward, as it would equal 3.
• When simplifying radical expressions, we can pull out pairs of values (or factors) as a single value outside the root.

This can help to simplify calculations and make the expression easier to work with, as we did with \( \sqrt{18} \), simplifying it to \(3 \sqrt{2}\).

Exponents

Exponents represent how many times a number is multiplied by itself. For instance, in \( a^5 \), the base \(a\) is multiplied by itself five times.

• When dealing with radicals, it's useful to express variables with exponents, so we can see which factors can be paired and brought out of the radical.
• For example, \(a^5\) can be broken into \(a^4 \cdot a = (a^2)^2 \cdot a\), allowing \(a^2\) to be brought outside as \(a^2\).

This method of simplifying helps to decrease the powers inside the radical and simplify the overall expression.

Variable Simplification

Simplifying variables inside a square root requires some attention to exponents. Here's the process: we split variables into parts with even exponents and those that aren’t. The aim is to make exponents as simple as possible outside the root.

• By breaking \(b^{17}\) into \(b^{16}\cdot b\), \(b^{16}\) can be simplified as \((b^8)^2\), bringing \(b^8\) outside.
• Similarly, \(a^5\) can be separated into \(a^4\cdot a\), which simplifies to \((a^2)^2\cdot a\), allowing \(a^2\) to come out of the radical.

Once simplified, these variables integrate with the numerical coefficients outside the square root, simplifying the expression neatly.