Simplify (3st)^2

Hey guys! Today, we're diving into a super common math topic that trips a lot of people up: simplifying algebraic expressions. Specifically, we're going to tackle this one: $(3 s t)^2$. Don't let the letters and the exponent scare you; it's actually pretty straightforward once you know the rules. We'll break it down step-by-step so you can nail these kinds of problems every time.

Understanding the Expression: $(3 s t)^2$

First off, let's look at what we've got here. We have a term inside the parentheses, $(3st)$, and it's being raised to the power of 2. This means we need to multiply the entire term by itself. So, $(3st)^2$ is the same as $(3st) \times (3st)$. This is the fundamental concept we'll be working with. When you have a term with multiple factors inside parentheses and an exponent outside, that exponent applies to every single factor inside. Think of it like a distribution rule for exponents. So, the '2' doesn't just apply to the 't'; it applies to the '3', the 's', and the 't'. This is a crucial point that often leads to mistakes. Many people might think it's just $3s t^2$ or something similar, but that's not correct because the exponent is outside the parentheses, indicating it affects everything within.

Applying the Power Rule

Now, let's apply that understanding. We need to square each component of the term $(3st)$. This means we have:

• The coefficient '3' needs to be squared: $3^2$
• The variable 's' needs to be squared: $s^2$
• The variable 't' needs to be squared: $t^2$

When we combine these squared components, we get $3^2 \times s^2 \times t^2$. This is where the simplification really happens. We know that $3^2$ is $3 \times 3$, which equals 9. So, the expression becomes $9 \times s^2 \times t^2$. We usually write this without the multiplication signs, just putting the terms next to each other. Therefore, our simplified expression is $9s^2t^2$.

Evaluating the Options

Let's look at the multiple-choice options provided to see which one matches our result. We have:

A. $6 s^2 t^2$ B. $9 s^2 t^2$ C. $3 st^2$ D. $2 s^2 t^2$

Comparing our calculated answer, $9s^2t^2$, with the options, we can clearly see that option B is the correct one. Option A is incorrect because it seems to have added the coefficients ($3+3=6$) instead of multiplying them, and it incorrectly squared the variables. Option C is incorrect because it only squared the 't' and didn't square the '3' or the 's' at all. Option D is incorrect for similar reasons to A, miscalculating the coefficient.

Why This Matters: The Power of Exponents

Understanding how exponents work, especially with multiple factors inside parentheses, is fundamental in algebra. This rule, often called the power of a product rule, states that $(ab)^n = a^n b^n$. In our case, we had $(3st)^2$, where $a=3$, $b=s$, and $c=t$ (if we consider it as $(abc)^n$). The rule extends to any number of factors: $(abc)^n = a^n b^n c^n$. This is why we had to apply the exponent '2' to the '3', the 's', and the 't' individually. If you forget this, you'll likely make errors in simplifying expressions, solving equations, and working with more complex algebraic structures later on. Mastering this basic exponent rule saves you a lot of headaches down the line!

Common Pitfalls and How to Avoid Them

So, what are the common mistakes beginners make with expressions like $(3st)^2$? The biggest one, as we touched upon, is forgetting to apply the exponent to all parts of the term inside the parentheses. Forgetting to square the coefficient (the '3' in this case) is super common. Someone might write $3s^2t^2$, which is wrong. Another mistake is confusing exponentiation with multiplication. They might see the '2' and think $3 \times 2 = 6$, leading to option A, $6s^2t^2$. Remember, squaring means multiplying the base by itself. So, $3^2 = 3 \times 3 = 9$, not $3 \times 2 = 6$. Also, ensure you're only applying the exponent to the correct base. In $(3st)^2$, the exponent '2' applies to the entire $(3st)$ term. If you had something like $3s^2t$, then the '2' only applies to the 's'. Always pay close attention to where the parentheses are and what the exponent is attached to.

To avoid these errors, the best strategy is to write out the steps explicitly when you're first learning. Instead of just jumping to the answer, write: $(3st)^2 = (3st) \times (3st)$. Then, group the like terms: $(3 \times 3) \times (s \times s) \times (t \times t)$. Finally, simplify each group: $9 \times s^2 \times t^2 = 9s^2t^2$. This methodical approach helps reinforce the rules and prevents careless mistakes. Also, practicing regularly is key! The more you see and solve these types of problems, the more natural the rules will become.

Conclusion: The Power of Precision

So, there you have it! Simplifying $(3st)^2$ boils down to applying the exponent to each factor within the parentheses. This gives us $3^2 \times s^2 \times t^2$, which equals $9s^2t^2$. This matches option B. Remember the power of a product rule: $(abc)^n = a^n b^n c^n$. Keep practicing these rules, pay attention to details like parentheses and exponents, and you'll be simplifying algebraic expressions like a pro in no time. Keep up the great work, mathletes!

Final Answer Check

We are asked to simplify the expression $(3st)^2$. This involves applying the exponent to each factor inside the parentheses. According to the rules of exponents, specifically the power of a product rule, $(xyz)^n = x^n y^n z^n$. Applying this rule to our expression, we get:

$(3st)^2 = 3^2 \times s^2 \times t^2$

Calculating the square of the coefficient:

$3^2 = 3 \times 3 = 9$

So, the expression becomes:

$9 \times s^2 \times t^2$

Which is written as $9s^2t^2$.

Now, let's compare this result with the given options:

A. $6 s^2 t^2$ B. $9 s^2 t^2$ C. $3 st^2$ D. $2 s^2 t^2$

Our simplified expression $9s^2t^2$ perfectly matches option B. Therefore, option B is the correct answer. It's crucial to remember that the exponent outside the parentheses applies to every factor inside, including the numerical coefficient. Overlooking this is a common error that leads to incorrect answers like option A, which seems to stem from incorrectly adding or multiplying the coefficient by the exponent ($3 \times 2 = 6$). Option C incorrectly applies the exponent, only affecting the 't', and ignoring the '3' and 's'. Option D seems to be a random incorrect coefficient. The mathematical precision required for these problems highlights the importance of understanding and correctly applying exponent rules. This fundamental concept is a building block for more complex algebraic manipulations, so ensuring a solid grasp here is vital for success in mathematics.