Simplify Radicals: Multiply √42r⁴s ⋅ √2s⁵

by Viktoria Ivanova 42 views

Hey guys! Let's dive into the world of simplifying radicals, specifically when we're dealing with multiplication and variables. We're going to tackle an expression that involves square roots, variables raised to different powers, and our ultimate goal is to express the answer in its simplest form. So, let's get started!

Understanding the Problem

The problem we're going to solve is:

42r4s⋅2s5\sqrt{42 r^4 s} \cdot \sqrt{2 s^5}

Where both r and s are greater than or equal to zero. This condition is crucial because we're dealing with square roots, and we need to ensure that we're not taking the square root of a negative number. Remember, the square root of a negative number is not a real number.

Simplifying radical expressions involves a few key steps, and we'll break them down one by one. Firstly, understanding the properties of radicals is very crucial. For example, we know that the square root of a product is equal to the product of the square roots. So, a∗b=a∗b\sqrt{a*b} = \sqrt{a} * \sqrt{b}. Secondly, we need to identify perfect squares within the radical. Thirdly, we extract those perfect squares. By following these steps, we can systematically reduce a radical expression to its simplest form. This involves identifying and extracting any perfect square factors from the radicand (the expression under the radical sign). This skill is incredibly useful not only in algebra but also in various fields of mathematics and science where radical expressions pop up frequently.

Step 1: Combining the Radicals

When you're multiplying radicals, a neat trick is that you can combine them under a single radical sign. It's like gathering all the troops before heading into battle! So, we can rewrite our expression as:

42r4s⋅2s5\sqrt{42 r^4 s \cdot 2 s^5}

This step makes the subsequent simplification process a lot easier. Think of it as consolidating the information. Instead of dealing with two separate square roots, we now have one big square root containing all the factors. This allows us to see the entire picture and identify common factors more easily. It is similar to combining like terms in an algebraic expression; it's about grouping things together to make them more manageable.

Now, let's simplify what's under the radical.

Step 2: Simplifying the Expression Under the Radical

Under the radical, we have the product 42 * r⁴ * s * 2 * s⁵. Let's simplify this by multiplying the constants and combining the variables:

  • Multiplying the constants: 42 * 2 = 84
  • Combining the 's' terms: s * s⁵ = s¹⁺⁵ = s⁶ (Remember the rule: when multiplying exponents with the same base, you add the exponents)

So, our expression under the radical now looks like:

84r4s6\sqrt{84 r^4 s^6}

Now, we have a much cleaner expression under the square root. This is progress! Breaking down the expression into its prime factors helps to reveal perfect squares, which can then be extracted from the radical. In this case, we've multiplied the constants and used the laws of exponents to combine the s terms. This step is crucial because it prepares the expression for the next stage: extracting the square roots. It is also a testament to the power of simplification in mathematics. By simplifying expressions, we make them easier to understand and manipulate, which is a fundamental principle in problem-solving.

Step 3: Factoring and Identifying Perfect Squares

Here's where the real fun begins! We need to factor 84 and see if we can find any perfect square factors. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.).

Let's break down 84 into its prime factors: 84 = 2 * 42 = 2 * 2 * 21 = 2 * 2 * 3 * 7 = 2² * 3 * 7

Now, let's rewrite our radical expression with the factored form of 84:

22imes3imes7imesr4imess6\sqrt{2^2 imes 3 imes 7 imes r^4 imes s^6}

Looking at this, we can identify some perfect squares:

  • 2² is a perfect square (2² = 4)
  • r⁴ is a perfect square because it can be written as (r²)²
  • s⁶ is a perfect square because it can be written as (s³)²

The ability to factor numbers and identify perfect squares is a fundamental skill in simplifying radicals. It's like detective work, where we're looking for clues hidden within the numbers and variables. Factoring a number into its prime factors is a powerful technique that can be applied in various areas of mathematics. In the context of radicals, it helps us to isolate the parts that can be simplified and the parts that need to stay under the radical sign. Recognizing perfect squares, like spotting familiar faces in a crowd, allows us to take those elements out of the square root, making the expression simpler and easier to work with.

Step 4: Extracting Perfect Squares

Now, let's take those perfect squares out from under the radical. Remember, the square root of a perfect square is just the base number. For example, $\sqrt{2^2} = 2$, $\sqrt{r^4} = r^2$, and $\sqrt{s^6} = s^3$.

So, we can rewrite our expression as:

2imesr2imess3imes3imes72 imes r^2 imes s^3 imes \sqrt{3 imes 7}

We took out the square root of 2², which is 2. We also extracted the square root of r⁴, which is r², and the square root of s⁶, which is s³. What's left under the radical are the factors that don't form perfect squares.

Step 5: Final Simplification

Finally, let's multiply the constants remaining under the radical (3 * 7 = 21) and write our answer in the simplest form:

2r2s3212 r^2 s^3 \sqrt{21}

And there we have it! We've successfully simplified the radical expression. This final step is like putting the finishing touches on a masterpiece. We've taken all the elements we've worked with – the numbers, the variables, the perfect squares – and assembled them into a final, simplified form. The result is a clear and concise expression that is much easier to understand and use than the original one. This is the essence of simplification: to take something complex and make it simple. It is not just about getting to the right answer, but also about presenting it in the most elegant and understandable way.

Final Answer

The simplest form of the expression 42r4s⋅2s5\sqrt{42 r^4 s} \cdot \sqrt{2 s^5} is:

2r2s321\boxed{2 r^2 s^3 \sqrt{21}}

So guys, that's how you multiply and simplify radicals! Remember to combine the radicals, simplify the expression under the radical, identify perfect squares, extract them, and then write your answer in the simplest form. Keep practicing, and you'll become a pro at this in no time! This process is like following a recipe in cooking. Each step is important, and when followed correctly, they lead to a delicious outcome. Similarly, in mathematics, each step in the simplification process is crucial, and when executed accurately, they lead to the simplified form of the expression. The more we practice, the more natural these steps become, and the more confident we become in our ability to tackle complex problems.