Simplify Radical Expressions: A Math Guide

Welcome to our guide on simplifying radical expressions! Today, we're going to dive deep into a common problem: simplifying expressions like $\sqrt{768 x^{19} y^{37}}$. Don't let those big numbers and exponents scare you; with a few key strategies, you'll be simplifying them like a pro. We'll break down the process step-by-step, explaining the logic behind each move. This skill is fundamental in algebra and pops up in various math contexts, so mastering it will definitely give you a boost in your mathematical journey. Let's get started on making these complex radicals much more manageable and easier to work with.

Understanding the Basics of Radical Simplification

To effectively tackle the simplification of radical expressions, it's crucial to understand the core principles. The main goal when simplifying a square root is to pull out as much as possible from under the radical sign. This involves finding perfect squares within the number and variables. For the numerical part, we look for the largest perfect square factor of the number under the radical. A perfect square is any number that results from squaring an integer (e.g., 4, 9, 16, 25, etc.). For instance, if we had $\sqrt{12}$, we'd recognize that 12 has a perfect square factor of 4 ($12 = 4 \times 3$). We can then rewrite $\sqrt{12}$ as $\sqrt{4 \times 3}$. Using the property of radicals that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we can separate this into $\sqrt{4} \times \sqrt{3}$. Since $\sqrt{4} = 2$, the simplified form becomes $2\sqrt{3}$.

When dealing with variables under the radical, like $\sqrt{x^{19}}$, we apply a similar logic, but with exponents. Remember that a square root is essentially an exponent of $1/2$. So, $\sqrt{x^{19}}$ is equivalent to $(x^{19})^{1/2} = x^{19/2}$. To simplify, we want to find the largest even exponent that is less than or equal to the given exponent. For $x^{19}$, the largest even exponent less than 19 is 18. We can rewrite $x^{19}$ as $x^{18} \times x^1$. Now, applying the radical property again, $\sqrt{x^{19}} = \sqrt{x^{18} \times x} = \sqrt{x^{18}} \times \sqrt{x}$. Since $\sqrt{x^{18}}$ simplifies to $x^{18/2} = x^9$, we have $x^9 \sqrt{x}$. The key takeaway here is that for any variable with an even exponent under a square root, say $x^n$ where $n$ is even, $\sqrt{x^n} = x^{n/2}$. If the exponent is odd, say $x^m$ where $m$ is odd, we split it into $\sqrt{x^{m-1} \times x}$, where $m-1$ is even, so $\sqrt{x^m} = \sqrt{x^{m-1}} \times \sqrt{x} = x^{(m-1)/2} \sqrt{x}$. This approach allows us to extract the maximum possible from under the radical.

Step-by-Step Simplification of $\sqrt{768 x^{19} y^{37}}$

Now, let's apply these principles to our specific problem: simplify $\sqrt{768 x^{19} y^{37}}$. We'll tackle the numerical part and the variable parts separately and then combine them. First, consider the number 768. We need to find the largest perfect square factor of 768. We can do this by prime factorization or by testing perfect squares. Let's try dividing by common perfect squares:

• $768 \div 4 = 192$
• $768 \div 9$ (doesn't divide evenly)
• $768 \div 16 = 48$
• $768 \div 25$ (doesn't divide evenly)
• $768 \div 36$ (doesn't divide evenly)
• $768 \div 64 = 12$
• $768 \div 144 = 5.33...$
• $768 \div 256 = 3$

So, $768 = 256 \times 3$. Since 256 is a perfect square ($\sqrt{256} = 16$), this is the largest perfect square factor we can easily identify. Thus, $\sqrt{768} = \sqrt{256 \times 3} = \sqrt{256} \times \sqrt{3} = 16\sqrt{3}$.

Next, let's handle the variables. We have $x^{19}$ and $y^{37}$ under the square root. For $x^{19}$, the largest even exponent less than 19 is 18. So, we write $x^{19} = x^{18} \times x^1$. Then, $\sqrt{x^{19}} = \sqrt{x^{18} \times x} = \sqrt{x^{18}} \times \sqrt{x} = x^{18/2} \sqrt{x} = x^9 \sqrt{x}$.

For $y^{37}$, the largest even exponent less than 37 is 36. So, we write $y^{37} = y^{36} \times y^1$. Then, $\sqrt{y^{37}} = \sqrt{y^{36} \times y} = \sqrt{y^{36}} \times \sqrt{y} = y^{36/2} \sqrt{y} = y^{18} \sqrt{y}$.

Now, we combine all the parts: the simplified numerical part, the simplified x part, and the simplified y part. We have $16\sqrt{3}$, $x^9\sqrt{x}$, and $y^{18}\sqrt{y}$.

Putting it all together, $\sqrt{768 x^{19} y^{37}} = \sqrt{768} \times \sqrt{x^{19}} \times \sqrt{y^{37}}$ $= (16\sqrt{3}) \times (x^9\sqrt{x}) \times (y^{18}\sqrt{y})$ $= 16 x^9 y^{18} \sqrt{3} \sqrt{x} \sqrt{y}$

Using the property $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$, we can combine the remaining square roots: $\, = 16 x^9 y^{18} \sqrt{3xy}$

Therefore, the expression equivalent to $\sqrt{768 x^{19} y^{37}}$ is $16 x^9 y^{18} \sqrt{3xy}$. This matches option A.

Analyzing the Options Provided

Let's take a moment to analyze the given options to see why only one is correct. We've already derived the correct answer as $16 x^9 y^{18} \sqrt{3xy}$. Now, let's look at each option and see how it compares.

A. $16 x^9 y^{18} \sqrt{3 x y}$

This option matches our derived answer exactly. The numerical coefficient 16 is the square root of the largest perfect square factor of 768 (which is 256). The variable $x^9$ is derived from $\sqrt{x^{18}}$ (the largest even power of x less than or equal to 19). The variable $y^{18}$ is derived from $\sqrt{y^{36}}$ (the largest even power of y less than or equal to 37). The remaining terms under the radical are $3xy$, which are the factors that couldn't be simplified further (3 from 768, x from $x^{19}$, and y from $y^{37}$). This option correctly simplifies the original expression.

B. $8 x^9 y^{18} \sqrt{12 x y}$

Let's see if this can be equivalent. The coefficients are different: 8 instead of 16. If we square 8, we get 64. $768 \div 64 = 12$. This means that $\sqrt{768}$ could be written as $\sqrt{64 \times 12} = 8\sqrt{12}$. The variable parts $x^9$ and $y^{18}$ are correct, as they represent the simplified parts of $\sqrt{x^{19}}$ and $\sqrt{y^{37}}$. The radical part $\sqrt{12xy}$ suggests that maybe 12 was left under the radical. However, 12 is not a prime number, and it has a perfect square factor of 4 ($12 = 4 \times 3$). If we were to simplify $\sqrt{12}$, it would become $\sqrt{4 \times 3} = 2\sqrt{3}$. So, if the expression were $8 x^9 y^{18} \sqrt{12xy}$, it could be further simplified by taking the $\sqrt{12}$ and simplifying it to $2\sqrt{3}$. This would lead to $8 x^9 y^{18} \times (2\sqrt{3}) \sqrt{xy} = 16 x^9 y^{18} \sqrt{3xy}$. Therefore, this option is not the fully simplified form.

C. $16 x^4 y^6 \sqrt{3 x^4 y}$

This option has the correct numerical coefficient (16), but the variable exponents are incorrect. We found that the simplified $x$ term should be $x^9$ and the simplified $y$ term should be $y^{18}$. Here, we see $x^4$ and $y^6$. If we were to square these and multiply by the terms under the radical, we would get $(x^4)^2 = x^8$ and $(y^6)^2 = y^{12}$. Combined with the $x^4 y$ under the radical, this would give us $x^8 \times x^4 y = x^{12} y$. This is far from the original $x^{19} y^{37}$. Thus, this option is incorrect due to incorrect variable simplification.

D. $8 x^4 y^6 \sqrt{12 x^4 y}$

This option is incorrect for multiple reasons. The numerical coefficient 8 is not the largest possible factor, as we found 16 works. The variable exponents $x^4$ and $y^6$ are also incorrect, as we determined they should be $x^9$ and $y^{18}$. Finally, the term $\sqrt{12x^4y}$ also contains a non-simplified radical part ($\sqrt{12}$ can be simplified to $2\sqrt{3}$) and an incorrect variable exponent for $x$ under the radical. This option is a combination of several simplification errors.

Conclusion: Mastering Radical Simplification

We've walked through the process of simplifying the radical expression $\sqrt{768 x^{19} y^{37}}$, arriving at the correct answer $16 x^9 y^{18} \sqrt{3xy}$. The key is to systematically break down the expression into its numerical and variable components. For the numerical part, always find the largest perfect square factor. For the variable parts, extract the largest even exponent that is less than or equal to the given exponent. Any remaining factors stay under the radical sign. This methodical approach ensures that you fully simplify the expression, just as option A demonstrates.

Key takeaways to remember when simplifying radicals:

1. Perfect Squares: Look for perfect square factors in the number. For $\sqrt{768}$, we found $768 = 256 \times 3$, where 256 is $16^2$.
2. Even Exponents: For variables under a square root, extract the largest possible even power. For $\sqrt{x^{19}}$, we take out $\sqrt{x^{18}} = x^9$. For $\sqrt{y^{37}}$, we take out $\sqrt{y^{36}} = y^{18}$.
3. Remainders: Whatever cannot be taken out (like the 3 from 768, the $x^1$ from $x^{19}$, and the $y^1$ from $y^{37}$) stays under the radical sign, forming $\sqrt{3xy}$.

By applying these rules consistently, you can confidently simplify any radical expression. Practice makes perfect, so try working through more examples on your own!

For further exploration and more complex examples of simplifying radicals, you can visit reputable math resources like Khan Academy or Math is Fun.