Equivalent Expression Of $\sqrt[2]{200k^{15}}$: A Math Guide

In this guide, we'll break down how to find the equivalent expression for $\sqrt[2]{200k^{15}}$, given that $k \neq 0$. This is a common type of problem in algebra that involves simplifying radicals and exponents. Let’s dive in and make it crystal clear!

Understanding the Problem

First, let's understand what the question is asking. We need to simplify the given expression, which involves a square root and a variable raised to a power. The key here is to factor the numbers and exponents inside the square root and then pull out any perfect squares. We'll take it step by step.

Breaking Down the Components

• The Square Root: The symbol $\sqrt[2]{}$ means we're looking for the square root, which is a number that, when multiplied by itself, gives us the number inside the root.
• The Number 200: We need to find the prime factorization of 200 to identify any perfect square factors.
• The Variable Term $k^{15}$: We need to simplify this by understanding how exponents work with radicals.
• The Condition $k \neq 0$: This condition is important because it tells us that $k$ cannot be zero, which prevents any undefined situations (like dividing by zero, which isn't applicable here but is a general caution in algebra).

Step-by-Step Simplification

Now, let's simplify the expression step by step:

Step 1: Factor 200

Let's find the prime factorization of 200. We can break it down as follows:

$200 = 2 imes 100$

$100 = 2 imes 50$

$50 = 2 imes 25$

$25 = 5 imes 5$

So, $200 = 2 imes 2 imes 2 imes 5 imes 5 = 2^3 imes 5^2$.

Step 2: Rewrite the Expression

Now, we can rewrite the original expression using this factorization:

$\sqrt[2]{200k^{15}} = \sqrt[2]{2^3 imes 5^2 imes k^{15}}$

Step 3: Simplify the Square Root

To simplify the square root, we look for pairs of factors (since it's a square root). We can rewrite the expression as:

$\sqrt[2]{2^2 imes 2 imes 5^2 imes k^{14} imes k}$

Here, we've separated $2^3$ into $2^2 imes 2$ and $k^{15}$ into $k^{14} imes k$. We did this because $2^2$, $5^2$, and $k^{14}$ are perfect squares.

Step 4: Take Out Perfect Squares

Now, we take the square roots of the perfect squares:

• $\sqrt[2]{2^2} = 2$
• $\sqrt[2]{5^2} = 5$
• $\sqrt[2]{k^{14}} = k^7$ (since $\sqrt[2]{k^{14}} = (k^{14})^{\frac{1}{2}} = k^{14 imes \frac{1}{2}} = k^7$)

So, we pull these out of the square root:

$2 imes 5 imes k^7 \sqrt[2]{2k}$

Step 5: Final Simplified Expression

Multiply the terms outside the square root:

$10k^7\sqrt[2]{2k}$

Common Mistakes to Avoid

When simplifying radicals, there are a few common mistakes to watch out for:

• Forgetting to Factor Completely: Always make sure you've factored the number inside the square root completely to find all perfect square factors.
• Incorrectly Simplifying Exponents: Remember the rules of exponents. When taking the square root of a variable raised to a power, divide the exponent by 2.
• Ignoring the Condition $k \neq 0$: While it didn't directly impact the simplification steps here, it's crucial to pay attention to these conditions in other problems.

Practice Problems

To solidify your understanding, try simplifying these expressions:

1. $\sqrt[2]{75x^9}$
2. $\sqrt[2]{162y^{12}}$
3. $\sqrt[2]{300z^{21}}$

Conclusion

Simplifying expressions with radicals and exponents might seem daunting at first, but by breaking it down step by step, it becomes much more manageable. Remember to factor completely, identify perfect squares, and apply the rules of exponents. Practice makes perfect, so keep at it!

By following this guide, you should now have a solid understanding of how to simplify expressions like $\sqrt[2]{200k^{15}}$. Remember, the key is to break down the problem into smaller, manageable steps. Factoring, identifying perfect squares, and applying exponent rules are your best friends in these situations.

Keep practicing, and you'll become a pro at simplifying radicals in no time! Feel free to revisit this guide whenever you need a refresher.

For more in-depth explanations and examples, you might find helpful resources on websites like Khan Academy. They offer a wealth of information and practice exercises to further enhance your understanding of algebra and radical expressions.

I hope this guide has been helpful! If you have any questions or need further clarification, don't hesitate to ask. Happy simplifying!