Simplifying Radicals: How To Simplify 3√135

Let's dive into simplifying the expression $3${sqrt{135}}$. It's a common task in mathematics, and mastering it can really boost your algebra skills. Radicals might seem intimidating at first, but with a step-by-step approach, you'll find they're quite manageable. So, let's break it down and make sure we understand each stage clearly.

Understanding the Basics of Simplifying Radicals

Before we tackle $3${sqrt{135}}$, let's quickly recap what it means to simplify a radical. Essentially, we want to express the number inside the square root (the radicand) in terms of its prime factors and pull out any perfect squares. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). The goal is to reduce the radicand to the smallest possible number while keeping the expression equivalent to the original. Remember, guys, simplifying radicals is all about making them easier to work with and understand.

For example, consider $\sqrt{20}$. We can rewrite 20 as $4 \times 5$, where 4 is a perfect square. So, $\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}$. This simplified form is much cleaner and easier to handle in further calculations.

Now, when we have a coefficient outside the radical, like in our case with $3\sqrt{135}$, we just keep that coefficient and multiply it by any factors we pull out of the square root. The coefficient stays put until we have something to combine it with. This is an important point to remember as we move forward.

Step-by-Step Simplification of $3\sqrt{135}$

Step 1: Prime Factorization of 135

The first thing we need to do is find the prime factorization of 135. This means breaking 135 down into its prime factors, which are prime numbers that multiply together to give 135. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11, etc.).

Let's start dividing 135 by the smallest prime number, 2. Since 135 is odd, it's not divisible by 2. Next, let's try dividing by 3. We find that $135 \div 3 = 45$. So, $135 = 3 \times 45$.

Now, we need to factor 45. Again, 45 is divisible by 3, and $45 \div 3 = 15$. So, $45 = 3 \times 15$, and $135 = 3 \times 3 \times 15$.

Finally, we factor 15. We know that $15 = 3 \times 5$. Thus, the prime factorization of 135 is $3 \times 3 \times 3 \times 5$, which can be written as $3^3 \times 5$.

Step 2: Rewriting the Radical

Now that we have the prime factorization of 135, we can rewrite the radical: $\sqrt{135} = \sqrt{3^3 \times 5}$. Remember, we are looking for perfect squares within the radical to simplify it. We can rewrite $3^3$ as $3^2 \times 3$, so we have $\sqrt{135} = \sqrt{3^2 \times 3 \times 5}$.

Step 3: Simplifying the Square Root

We can now simplify the square root by taking out the perfect square. Since $\sqrt{3^2} = 3$, we can pull the 3 out of the square root: $\sqrt{3^2 \times 3 \times 5} = 3\sqrt{3 \times 5} = 3\sqrt{15}$.

Step 4: Multiplying by the Coefficient

Finally, we need to multiply this simplified radical by the coefficient that was originally outside the radical, which is 3. So, we have $3 \times (3\sqrt{15}) = 9\sqrt{15}$. Therefore, the simplified form of $3\sqrt{135}$ is $9\sqrt{15}$.

Common Mistakes to Avoid

When simplifying radicals, there are a few common mistakes that you should watch out for:

1. Forgetting to Multiply the Coefficient: Always remember to multiply the coefficient outside the radical by any factors you pull out of the square root.
2. Not Fully Simplifying: Make sure you've pulled out all possible perfect squares from the radicand. Sometimes, you might need to factor the radicand multiple times to find all the perfect squares.
3. Incorrect Prime Factorization: Double-check your prime factorization to make sure you haven't made any errors. A mistake in the prime factorization will lead to an incorrect simplified radical.
4. Confusing Multiplication and Addition: Be careful not to confuse multiplying the coefficient with adding it. The coefficient is always multiplied by the simplified radical.

Practice Problems

To solidify your understanding, here are a few practice problems you can try:

1. Simplify $2\sqrt{72}$
2. Simplify $5\sqrt{48}$
3. Simplify $4\sqrt{125}$

Solutions to Practice Problems

1. $2\sqrt{72} = 2\sqrt{36 \times 2} = 2 \times 6\sqrt{2} = 12\sqrt{2}$
2. $5\sqrt{48} = 5\sqrt{16 \times 3} = 5 \times 4\sqrt{3} = 20\sqrt{3}$
3. $4\sqrt{125} = 4\sqrt{25 \times 5} = 4 \times 5\sqrt{5} = 20\sqrt{5}$

Conclusion

Simplifying radicals is a fundamental skill in algebra, and with practice, you'll become more comfortable with it. Remember to break down the radicand into its prime factors, identify any perfect squares, and pull them out of the square root. And, of course, don't forget to multiply by the coefficient! With these tips in mind, you'll be simplifying radicals like a pro in no time. Keep practicing, guys, and you'll master it!