Simplify Radical Expressions: $\sqrt{2 S^3 T^5} \cdot \sqrt{42 S^3 T}$

Hey guys! Today, we're diving deep into the awesome world of mathematics, specifically tackling how to multiply radical expressions. This might sound a bit intimidating at first, but trust me, once you get the hang of the rules, it's a piece of cake! We're going to break down the expression $\sqrt{2 s^3 t^5} \cdot \sqrt{42 s^3 t}$, assuming that our variables $s$ and $t$ are non-negative. This assumption is super important because it means we don't have to worry about dealing with imaginary numbers when we take the square root of variables raised to odd powers. So, let's get this party started and simplify this expression step-by-step.

Understanding the Basics of Radical Multiplication

Before we jump into our specific problem, let's quickly refresh some fundamental rules about multiplying radicals. The golden rule here is that if you have the square root of one number multiplied by the square root of another number, you can combine them under a single square root sign. Mathematically, this is expressed as $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. This property is our best friend when dealing with multiplication problems like the one we have. It allows us to merge the two separate radical expressions into one larger one, which we can then work on simplifying. Think of it as gathering all your components under one roof before you start tidying up. This simplification process often involves pulling out any perfect squares from under the radical sign. A perfect square is any number or variable expression that can be obtained by squaring another number or expression. For instance, 9 is a perfect square ($3^2$), and $x^4$ is a perfect square ($(x^2)^2$). The key to simplifying radicals is to identify these perfect squares within the radicand (the expression under the radical sign) and then take their square root, effectively removing them from the radical. We'll be using this technique extensively as we move forward.

Step-by-Step Simplification

Alright, team, let's get down to business with our expression: $\sqrt{2 s^3 t^5} \cdot \sqrt{42 s^3 t}$.

First off, we're going to use that handy property of radicals we just discussed: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. So, we combine our two expressions under a single square root:

$\sqrt{(2 s^3 t^5) \cdot (42 s^3 t)}$

Now, let's multiply the terms inside the radical. We'll multiply the numerical coefficients and then combine the variables with the same base by adding their exponents:

• Coefficients: $2 \cdot 42 = 84$
• 's' variables: $s^3 \cdot s^3 = s^{3+3} = s^6$
• 't' variables: $t^5 \cdot t = t^{5+1} = t^6$

Putting it all together, our expression under the radical becomes:

$\sqrt{84 s^6 t^6}$

So far, so good, right? We've successfully merged our two radicals into one. Now comes the part where we simplify this beast by looking for perfect squares within the radicand $84 s^6 t^6$.

Factoring and Identifying Perfect Squares

Our goal now is to simplify $\sqrt{84 s^6 t^6}$. To do this, we need to break down the number 84 and the variable terms $s^6$ and $t^6$ into their factors, specifically looking for factors that are perfect squares. Remember, a perfect square has an exponent that is an even number. This is because when you take the square root of a variable raised to an even power, you simply divide the exponent by 2.

Let's start with the numerical coefficient, 84. We need to find the largest perfect square that divides 84. Let's list some perfect squares: $1, 4, 9, 16, 25, 36, 49, 64, ...$

• Is 84 divisible by 4? Yes, $84 = 4 \cdot 21$. 4 is a perfect square ($2^2$).
• Is 84 divisible by 9? No.
• Is 84 divisible by 16? No.
• Is 84 divisible by 25? No.
• Is 84 divisible by 36? No.

So, the largest perfect square factor of 84 is 4. We can rewrite 84 as $4 \cdot 21$.

Now let's look at the variable terms:

• $s^6$: The exponent is 6, which is an even number. This means $s^6$ is already a perfect square. We can write it as $(s^3)^2$. So, $\sqrt{s^6} = s^{6/2} = s^3$.
• $t^6$: Similarly, the exponent 6 is even, so $t^6$ is a perfect square. We can write it as $(t^3)^2$. Thus, $\sqrt{t^6} = t^{6/2} = t^3$.

Now we can rewrite our expression inside the radical using these perfect square factors:

$\sqrt{84 s^6 t^6} = \sqrt{(4 \cdot 21) \cdot s^6 \cdot t^6}$

We can rearrange this to group the perfect squares together:

$\sqrt{(4 \cdot s^6 \cdot t^6) \cdot 21}$

Extracting Perfect Squares from the Radical

This is the fun part, guys! We get to pull those perfect squares out from under the radical sign. Remember, the square root of a perfect square is just the base itself. So, when we take the square root of $4 s^6 t^6$, we get:

• $\sqrt{4} = 2$
• $\sqrt{s^6} = s^3$
• $\sqrt{t^6} = t^3$

Combining these, we get $2 s^3 t^3$. This part is now outside the radical.

What's left inside the radical is the factor 21, which has no more perfect square factors (other than 1, which doesn't change anything).

So, after extracting the perfect squares, our expression transforms into:

$2 s^3 t^3 \sqrt{21}$

This is our simplified form! We've taken the original, more complex expression $\sqrt{2 s^3 t^5} \cdot \sqrt{42 s^3 t}$ and reduced it to its simplest terms. The key steps were combining the radicals using the multiplication property, multiplying the terms inside, finding the perfect square factors of the coefficient and variables, and then extracting those perfect squares from the radical.

Final Answer and Verification

So, the final simplified answer to multiplying $\sqrt{2 s^3 t^5} \cdot \sqrt{42 s^3 t}$ is $2 s^3 t^3 \sqrt{21}$. We confirmed this by breaking down each part of the expression, applying the rules of radical multiplication, and then systematically simplifying by factoring out perfect squares. The assumption that $s$ and $t$ are non-negative was crucial here, as it allowed us to take the square root of expressions like $s^6$ and $t^6$ without any ambiguity. If $s$ or $t$ could be negative, we would need to use absolute value signs around the variables that came out of the radical to ensure the result remained non-negative, as a square root symbol by convention denotes the principal (non-negative) root.

Let's do a quick mental check. We started with two radicals, and ended up with one term outside the radical and one term inside. The term outside, $2 s^3 t^3$, represents the 'perfect' part we could pull out, and the term inside, $\sqrt{21}$, is the 'remainder' that couldn't be simplified further. This structure is typical for simplified radical expressions.

Key Takeaways:

1. Product Property of Radicals: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. Use this to combine radicals.
2. Combine Like Terms: When multiplying, multiply coefficients and add exponents of variables with the same base.
3. Factor Radicands: Break down the number and variable parts of the expression under the radical into factors, looking for perfect squares.
4. Extract Perfect Squares: Take the square root of perfect square factors and move them outside the radical.
5. Simplify Remaining Radical: Ensure the expression left under the radical has no perfect square factors.

Keep practicing these steps, and you'll be a radical simplification pro in no time! Math is all about understanding these fundamental rules and applying them logically. Don't be afraid to break down complex problems into smaller, manageable steps. You've got this!