Simplify Radical Expression Equivalent To √(55x^7y^6 / 11x^11y^8)

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Hey guys! Let's dive into a cool math problem today. We're going to break down the expression $\sqrt{\frac{55 x^7 y^6}{11 x^{11} y^8}}$ and find out which of the given options is its equivalent. This involves simplifying radicals and using the rules of exponents. Don't worry, we'll take it step by step so it’s super clear.

Simplifying the Expression

First things first, let's simplify the fraction inside the square root. We have $\frac{55 x^7 y^6}{11 x^{11} y^8}$. We can tackle the coefficients and the variables separately. So, when dealing with exponents, it's like a superhero power – we can make complex stuff look super simple! Remember, if you are dividing powers with the same base, you subtract the exponents. It's like magic, but it's math!

Diving into the Coefficients

The coefficients are 55 and 11. What's 55 divided by 11? It’s 5! So, we’ve got that part sorted. It's always satisfying to simplify the numbers first, right? It feels like we're already making progress.

Taming the Variables

Now for the variables. We have $x^7$ in the numerator and $x^{11}$ in the denominator. When we divide these, we subtract the exponents: $7 - 11 = -4$. So we have $x^{-4}$. Remember, a negative exponent means we can move the variable to the denominator and make the exponent positive. It’s like flipping the script, but with algebra! Similarly, for $y$, we have $y^6$ in the numerator and $y^8$ in the denominator. Subtracting the exponents gives us $6 - 8 = -2$. So we get $y^{-2}$. Again, this means we can move $y^2$ to the denominator.

Putting It All Together

So, after simplifying the fraction, we have $\frac{5}{x^4 y^2}$ inside the square root. Our expression now looks like $\sqrt{\frac{5}{x^4 y^2}}$. See how much cleaner that looks? It's like we've decluttered our math space!

Tackling the Square Root

Now comes the fun part: taking the square root. Remember, the square root of a fraction is the square root of the numerator divided by the square root of the denominator. It's like giving each part its own mini square root party!

Numerator: Square Root of 5

The square root of 5 is just $\sqrt{5}$ because 5 is a prime number. We can't simplify it further, so it stays as $\sqrt{5}$. Sometimes, the simplest answers are the coolest ones.

Denominator: Square Root of $x^4 y^2$

For the denominator, we have $\sqrt{x^4 y^2}$. Taking the square root of $x^4$ means dividing the exponent by 2, so we get $x^2$. Similarly, the square root of $y^2$ is just $y$. So the square root of the denominator simplifies to $x^2 y$. It's like we're peeling away layers to reveal the simple truth underneath!

The Final Simplified Expression

Putting it all together, our simplified expression is $\frac{\sqrt{5}}{x^2 y}$. This is the final form, and it matches one of the options given. High five! We nailed it.

Comparing with the Options

Let's quickly compare our result with the given options:

• A. $\frac{x^2 \sqrt{5}}{y}$
• B. $\frac{y \sqrt{5}}{x^2}$
• C. $\frac{\sqrt{5}}{x^2 y}$
• D. $\frac{x \sqrt{5}}{y}$

Our simplified expression, $\frac{\sqrt{5}}{x^2 y}$, matches option C. So, we’ve confirmed our answer. It’s like finding the missing piece of a puzzle and everything clicks into place!

Why Other Options Are Incorrect

It's always good to understand why the other options don't work. This helps us avoid making similar mistakes in the future. Let's break it down:

• Option A, $\frac{x^2 \sqrt{5}}{y}$, has the $x^2$ term in the numerator instead of the denominator. This is the reverse of what we found when simplifying the variables. It’s like having the volume up when you meant to turn it down – a little mistake can make a big difference!
• Option B, $\frac{y \sqrt{5}}{x^2}$, has the $y$ term in the numerator, which is also incorrect. Remember, $y$ was in the denominator after simplifying the initial fraction. It's like putting the cart before the horse – the order matters!
• Option D, $\frac{x \sqrt{5}}{y}$, has $x$ in the numerator, which is incorrect. We found $x^2$ in the denominator. It's like getting the ingredients right but mixing them in the wrong proportions – the outcome won't be quite right.

Final Answer

So, the correct answer is C. $\frac{\sqrt{5}}{x^2 y}$. We've successfully simplified the expression and found its equivalent form. Math can be like a detective game, where we follow clues to find the solution. And when we get it right, it's super rewarding!

Rules of Exponents

To really nail these types of problems, let's do a quick recap on the rules of exponents. These rules are your best friends when simplifying expressions. Think of them as your math toolbox – always handy to have around!

Product of Powers

When multiplying powers with the same base, you add the exponents: $a^m * a^n = a^{m+n}$. This is like inviting more friends to the party – the more, the merrier, and we just add them up!

Quotient of Powers

When dividing powers with the same base, you subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$. We used this one extensively in our problem. It’s like taking away some of the partygoers – we subtract them to see who's left.

Power of a Power

When raising a power to another power, you multiply the exponents: $(a^m)^n = a^{mn}$. This is like having a party within a party – we multiply the exponent to see the total effect.

Power of a Product

The power of a product is the product of the powers: $(ab)^n = a^n b^n$. This rule is super useful when dealing with multiple variables inside parentheses. It's like giving each guest at the party their own special treat!

Power of a Quotient

The power of a quotient is the quotient of the powers: $(\frac{a}{b})^n = \frac{a^n}{b^n}$. This is similar to the power of a product but for division. It's like dividing the treats equally among the guests.

Negative Exponent

A negative exponent means you take the reciprocal of the base raised to the positive exponent: $a^{-n} = \frac{1}{a^n}$. We used this one too! It’s like turning things upside down to see a new perspective.

Zero Exponent

Any number raised to the power of 0 is 1: $a^0 = 1$. This is a fun one to remember – anything to the power of zero becomes one. It’s like magic!

Fractional Exponents

A fractional exponent represents a radical: $a^{\frac{1}{n}} = \sqrt[n]{a}$. This connects exponents to radicals, which is super useful for simplifying expressions like ours. It's like uncovering a hidden connection between two seemingly different things.

Square Roots and Radicals

Let's also touch on square roots and radicals. These are closely related to exponents, especially fractional exponents. Understanding radicals is crucial for simplifying expressions like the one we tackled.

Basics of Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because $5 * 5 = 25$. It's like finding the twin of a number that, when paired together, makes the original number.

Simplifying Square Roots

To simplify a square root, you look for perfect square factors within the number. For example, to simplify $\sqrt{48}$, you can break 48 down into $16 * 3$, where 16 is a perfect square. So, $\sqrt{48} = \sqrt{16 * 3} = \sqrt{16} * \sqrt{3} = 4\sqrt{3}$. It's like breaking a big problem into smaller, more manageable parts.

Radicals with Variables

When dealing with variables under a square root, you divide the exponent by 2. For example, $\sqrt{x^6} = x^3$ because $6 / 2 = 3$. This is just like our earlier example with $x^4$ and $y^2$. It’s like sharing the exponent equally between the twins!

Rationalizing the Denominator

Sometimes, you might end up with a square root in the denominator of a fraction. To get rid of it, you can rationalize the denominator by multiplying both the numerator and denominator by the square root. For example, to rationalize $\frac{1}{\sqrt{2}}$, you multiply both the numerator and denominator by $\sqrt{2}$ to get $\frac{\sqrt{2}}{2}$. It’s like making sure the house is tidy by getting rid of any unwanted guests (in this case, the square root in the denominator).

Practice Makes Perfect

The best way to get comfortable with these concepts is practice. Try working through similar problems, and don't be afraid to make mistakes. Mistakes are just learning opportunities in disguise! The more you practice, the more these rules and techniques will become second nature. It's like learning to ride a bike – it might seem wobbly at first, but with practice, you'll be cruising along smoothly in no time!

So, there you have it! We've successfully simplified a complex expression, understood the rules of exponents and radicals, and compared our answer with the given options. Keep practicing, and you'll become a math whiz in no time!