The Simplification of 27⁴ʹ³ and Its Relationship with Cube Roots

What is 27⁴ʹ³ simplified to in terms of a cube root?

• A. ³√(27²)
• B. ³√(27⁴) = (³√27)²
• C. ³√(27⁴) = (³√27)⁴ = 3⁴ = 81
• D. 81

Answer: (C)

To understand why the choice is correct, it is essential to break down the expression 27⁴ʹ³ in terms of cube roots. Start by recalling that 27 can be expressed as 3³, which means that 27⁴ can be rewritten in its prime factor form: 27⁴ = (3³)⁴ = 3¹². Now, when we simplify 27⁴ under a cube root, we can use the property of roots: ³√(27⁴) can be rewritten as ³√(3¹²). Utilizing the property of exponents, we divide the exponent by the root: ³√(3¹²) = 3^(12/3) = 3⁴. Recognizing that 3⁴ equals 81 confirms that the expression ³√(27⁴) simplifies to 81. This step-by-step process is why this choice is the correct option. It accurately reflects the logarithmic and exponential relationships involved in dealing with cube roots and powers. The answer correctly arrives at 81 through its acknowledgment of the cube root’s properties and the arithmetic involved with exponents.

Have you ever looked at an equation, thrown your hands up, and thought, "What on earth does this even mean?" Well, let’s change that feeling with a more relatable approach to simplifying (27^{4 \div 3}). Grab your pencil, and let’s dig into it!

First off, remember that 27 isn’t just a number; it’s also a hidden treasure chest of cubes. You might know that 27 can be rewritten as (3^3). If you didn’t, now’s the time to remember that juicy nugget! So, when we’re faced with (27^{4 \div 3}), we can plug in that value.

Here’s what we get when we rewrite the expression:

[

27^{4} = (3^{3})^{4} = 3^{12}

]

Pretty straightforward, right? Now, let’s consider how we simplify this under a cube root. You might have encountered cube roots before, but let’s recap: the cube root essentially looks for a number that, when multiplied by itself three times, gives you the original number.

So now, we move on to our cube root transformation:

[

\sqrt[3]{27^{4}} \text{ can be rewritten as } \sqrt[3]{3^{12}}

]

You see where we’re going? Utilizing our knowledge of exponents, we divide that 12 by the root, 3:

[

\sqrt[3]{3^{12}} = 3^{(12/3)} = 3^{4}

]

And voilà! If you crunch those numbers, (3^{4}) equals 81.

Why does it matter? Understanding concepts like this isn’t just about passing tests; it’s about arming yourself with critical thinking skills that will carry you through your academic journey and beyond.

But hold on a minute! Let's get back to what this means for you in your studies. Think of it this way: every time you encounter a similar expression or concept, you’re building a puzzle. Each component, whether it’s simplifying, multiplying, or even factoring, helps create a clearer picture of mathematics as a whole.

This breakdown of (27^{4 \div 3} = (27^{4})) shows how powerful it can be to understand the properties of exponents and roots. You don’t just memorize rules; you learn the logic behind them—because at the end of the day, math is about relationships, whether it's between numbers or broader concepts.

So the next time you come across a tricky problem, remember: it’s all about piecing it together bit by bit, inviting those higher-order thinking skills into the fold.

Keep practicing. After all, understanding is just a cube root away!