Simplifying And Comparing Radical Expressions: A Step-by-Step Guide

Hey guys! Ever stumbled upon a math problem that looks like it's speaking a different language? Those radical expressions can seem intimidating, but trust me, they're not as scary as they look. Let's break down a problem that involves simplifying and comparing these expressions. We'll take it one step at a time, so you'll be a pro in no time!

The Challenge: Simplifying and Comparing Radicals

We've got a few radical expressions here that we need to understand and compare. The expressions are:

1. $11 \sqrt{45} - 4 \sqrt{5}$
2. $7 \sqrt{40}$
3. $14 \sqrt{10}$
4. $29 \sqrt{5}$
5. $95 \sqrt{5}$

Our mission, should we choose to accept it (and we do!), is to simplify these expressions and figure out how they relate to each other. This involves a bit of radical simplification, some arithmetic, and a dash of algebraic manipulation. So, let's dive in!

Step 1: Simplifying the First Expression ($11 \sqrt{45} - 4 \sqrt{5}$)

Our first task is to simplify $11 \sqrt{45} - 4 \sqrt{5}$. The key here is to look for perfect square factors within the radicals. Remember, a perfect square is a number that can be obtained by squaring an integer (like 4, 9, 16, etc.).

Breaking Down $\sqrt{45}$

First, let's focus on $\sqrt{45}$. Can we find a perfect square that divides 45? Absolutely! 45 can be written as $9 \times 5$, and 9 is a perfect square ($3^2 = 9$). So, we can rewrite $\sqrt{45}$ as $\sqrt{9 \times 5}$.

Using the property $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we get:

$\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$

Substituting Back into the Expression

Now, let's substitute this back into our original expression:

$11 \sqrt{45} - 4 \sqrt{5} = 11(3\sqrt{5}) - 4\sqrt{5}$

Simplifying Further

Next, we multiply:

$11(3\sqrt{5}) = 33\sqrt{5}$

So, our expression becomes:

$33\sqrt{5} - 4\sqrt{5}$

Now we can combine these like terms (since they both have $\sqrt{5}$):

$33\sqrt{5} - 4\sqrt{5} = (33 - 4)\sqrt{5} = 29\sqrt{5}$

Boom! The simplified form of $11 \sqrt{45} - 4 \sqrt{5}$ is $29\sqrt{5}$.

Key Takeaway

The trick here is to identify perfect square factors within the radical. This allows us to simplify the expression and make it easier to compare with other radicals.

Step 2: Simplifying the Second Expression ($7 \sqrt{40}$)

Now let's tackle the second expression: $7 \sqrt{40}$. Again, we're on the hunt for perfect square factors within the radical.

Breaking Down $\sqrt{40}$

Think about the factors of 40. Can we find a perfect square? Yes! 40 can be written as $4 \times 10$, and 4 is a perfect square ($2^2 = 4$). So, we can rewrite $\sqrt{40}$ as $\sqrt{4 \times 10}$.

Using the property $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we get:

$\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}$

Substituting Back into the Expression

Now, let's substitute this back into our original expression:

$7 \sqrt{40} = 7(2\sqrt{10})$

Simplifying Further

Next, we multiply:

$7(2\sqrt{10}) = 14\sqrt{10}$

Ta-da! The simplified form of $7 \sqrt{40}$ is $14\sqrt{10}$.

Key Takeaway

Remember, always look for the largest perfect square that divides the number under the radical. This will simplify your work and give you the most reduced form directly.

Step 3: Analyzing the Remaining Expressions

Now let's take a look at the remaining expressions. Some of them are already in a simplified form, which is great news for us!

1. $14 \sqrt{10}$ (already simplified)
2. $29 \sqrt{5}$ (already simplified)
3. $95 \sqrt{5}$ (already simplified)

Notice that $14\sqrt{10}$, $29\sqrt{5}$, and $95\sqrt{5}$ are already in their simplest forms. This means we don't need to do any further simplification for these. They're ready to be compared!

Step 4: Comparing the Simplified Expressions

Now that we've simplified all the expressions, let's put them all together and see what we've got:

1. $11 \sqrt{45} - 4 \sqrt{5} = 29\sqrt{5}$
2. $7 \sqrt{40} = 14\sqrt{10}$
3. $14 \sqrt{10}$
4. $29 \sqrt{5}$
5. $95 \sqrt{5}$

Now we can easily compare them. Notice that we have two expressions with $\sqrt{10}$ and three expressions with $\sqrt{5}$.

Comparing Expressions with $\sqrt{5}$

Let's compare the expressions with $\sqrt{5}$ first:

• $29\sqrt{5}$
• $29\sqrt{5}$
• $95\sqrt{5}$

It's clear that $95\sqrt{5}$ is the largest among these, and the other two are equal.

Comparing Expressions with $\sqrt{10}$

Now let's look at the expressions with $\sqrt{10}$:

• $14\sqrt{10}$
• $14\sqrt{10}$

These two are equal.

Key Takeaway

When comparing radical expressions, make sure the radicals are in their simplest form. This allows you to easily compare the coefficients (the numbers in front of the radicals) and determine the relative sizes of the expressions.

Step 5: Final Analysis and Insights

Alright, let's bring it all together! We've simplified and compared our expressions. Here's what we've found:

• $11 \sqrt{45} - 4 \sqrt{5} = 29\sqrt{5}$
• $7 \sqrt{40} = 14\sqrt{10}$
• $14 \sqrt{10}$
• $29 \sqrt{5}$
• $95 \sqrt{5}$

Observations:

• We have two pairs of equal expressions: $11 \sqrt{45} - 4 \sqrt{5}$ and $29 \sqrt{5}$ are equal, and the two $14 \sqrt{10}$ expressions are equal.
• $95 \sqrt{5}$ is the largest expression.

Comparing $\sqrt{5}$ and $\sqrt{10}$

One interesting point to note is the difference between expressions with $\sqrt{5}$ and $\sqrt{10}$. Since 10 is greater than 5, $\sqrt{10}$ is greater than $\sqrt{5}$. This means that even though the coefficient in front of $\sqrt{5}$ might be larger in some cases, the $\sqrt{10}$ expressions could still be greater depending on the coefficients.

Final Thoughts

Simplifying and comparing radical expressions might seem tricky at first, but with a systematic approach, it becomes much more manageable. The key steps are:

1. Identify and factor out perfect squares from the radicals.
2. Simplify the expressions.
3. Compare the coefficients of like radicals.

By following these steps, you can confidently tackle any radical expression problem that comes your way!

Final Words

So, there you have it! We've successfully decoded these radical expressions. Remember, the world of math is like a puzzle, and each step we take brings us closer to the solution. Keep practicing, and you'll become a radical master in no time!