Simplify Square Roots: $\sqrt{90 B^2 C^4}$

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the awesome world of algebra to tackle a fun little problem: simplifying the square root of $90 b^2 c^4$. Now, I know sometimes math can look a bit intimidating, especially with all those symbols and variables floating around, but trust me, once you break it down, it's totally manageable and even kind of satisfying. We're going to walk through this step-by-step, making sure you guys understand every part of the process. Our main goal here is to simplify the expression $\sqrt{90 b^2 c^4}$ by pulling out any perfect squares from under the radical sign. Think of it like this: we're trying to get as much 'stuff' out of the house (the square root) as possible, leaving only the irreducible bits behind. This technique is super handy in all sorts of math problems, from solving equations to graphing functions, so mastering it is a major win. We'll be looking for perfect square factors within the number $90$ and within the variable terms $b^2$ and $c^4$. Remember, a square root asks the question, 'What number, when multiplied by itself, gives you the number inside?' For example, the square root of $9$ is $3$ because $3 \times 3 = 9$. When we deal with variables, the same principle applies. The square root of $b^2$ is $b$ because $b \times b = b^2$, and the square root of $c^4$ is $c^2$ because $c^2 \times c^2 = c^4$. So, let's get our hands dirty and simplify $\sqrt{90 b^2 c^4}$!

Breaking Down the Expression: Finding Perfect Squares

Alright team, let's get down to business with our expression: $\sqrt{90 b^2 c^4}$. The first thing we want to do is look at the number part, $90$, and see if we can break it down into factors, specifically looking for perfect squares. A perfect square is any number that's the result of squaring an integer (like $1, 4, 9, 16, 25,$ etc.). So, let's find the prime factorization of $90$. We can see that $90 = 9 \times 10$. And hey, $9$ is a perfect square! That's awesome. Now let's break down $10$: $10 = 2 \times 5$. Neither $2$ nor $5$ are perfect squares, and they are prime numbers, so we can't break them down any further. So, the prime factorization of $90$ is $2 \times 3 \times 3 \times 5$, or $2 \times 3^2 \times 5$. The important part here is that we've identified $3^2$ as a perfect square factor within $90$. This means we can rewrite $90$ as $9 \times 10$, where $9$ is our perfect square.

Now, let's turn our attention to the variable parts: $b^2$ and $c^4$. These are already pretty friendly to square roots. Remember, the square root operation is the inverse of squaring. So, when we see $b^2$ under a square root, we can simplify it directly. The square root of $b^2$ is simply $b$ (assuming $b$ is non-negative, which is a common convention when dealing with basic simplification problems like this, otherwise we'd need absolute value bars, $|b|$). Similarly, for $c^4$, we need to think, 'What do we square to get $c^4$?' That would be $c^2$, because $(c^2)^2 = c^2 \times c^2 = c^4$. So, the square root of $c^4$ is $c^2$. This means both the $b^2$ and the $c^4$ terms under the square root are perfect squares that we can pull out.

So, to recap, we've identified:

• $90$ can be written as $9 \times 10$, where $9$ is a perfect square ($3^2$).
• $b^2$ is a perfect square, and its square root is $b$.
• $c^4$ is a perfect square, and its square root is $c^2$.

Our mission to simplify the square root $\sqrt{90 b^2 c^4}$ is well underway, and we've successfully identified all the perfect square components within the expression. This breakdown is crucial because the rule for simplifying square roots is to extract the square root of any perfect square factor. Stay tuned for the next step where we'll put it all together and get our final simplified answer!

Extracting Perfect Squares: The Simplification Process

Now that we've identified all the perfect square factors within $\sqrt{90 b^2 c^4}$, it's time for the exciting part: pulling them out! Remember our rule: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$. We can use this property to separate the perfect squares from the non-perfect squares under the radical.

Let's rewrite our expression using the perfect square factors we found. We established that $90 = 9 \times 10$, $b^2$ is $b^2$, and $c^4$ is $c^4$. So, we can rewrite the expression inside the square root as: $90 b^2 c^4 = 9 \times 10 \times b^2 \times c^4$.

Now, let's apply the square root to each of these factors:

$\sqrt{90 b^2 c^4} = \sqrt{9 \times 10 \times b^2 \times c^4}$

Using the property $\sqrt{a \times b \times c \times d} = \sqrt{a} \times \sqrt{b} \times \sqrt{c} \times \sqrt{d}$, we can separate this into:

$\sqrt{9} \times \sqrt{10} \times \sqrt{b^2} \times \sqrt{c^4}$

Let's evaluate each of these individually:

• $\\sqrt{9}$: As we discussed, $9$ is a perfect square ($3^2$), so its square root is $3$.
• $\\sqrt{10}$: The number $10$ has prime factors $2$ and $5$, neither of which is a perfect square. So, $\sqrt{10}$ cannot be simplified further. It stays under the radical.
• $\\sqrt{b^2}$: The square root of $b^2$ is $b$. (Again, assuming $b \ge 0$ for simplicity. If $b$ could be negative, we'd use $|b|$).
• $\\sqrt{c^4}$: The square root of $c^4$ is $c^2$, because $(c^2)^2 = c^4$.

Now, let's put it all back together. We multiply the terms that we took out of the square root:

$3 \times b \times c^2 \times \sqrt{10}$

This simplifies to:

$3bc^2\sqrt{10}$

And there you have it! We've successfully simplified the square root $\sqrt{90 b^2 c^4}$. The term $\sqrt{10}$ is now considered the