Simplify This Cube Root Expression

Hello math enthusiasts! Today, we're diving into the fascinating world of algebraic simplification, specifically tackling an expression involving cube roots. Have you ever looked at a complex mathematical expression and wondered how to make it simpler, more elegant, and easier to understand? That's precisely what we're going to do with the following sum: $3 b^2(\sqrt[3]{54 a})+3\left(\sqrt[3]{2 a b^6}\right)$. This problem is a fantastic way to practice your skills in manipulating radicals, understanding exponents, and combining like terms. We'll break it down step-by-step, ensuring that by the end, you'll not only know the answer but also the why behind it. So, grab your thinking caps, and let's embark on this mathematical journey together to simplify this expression and arrive at one of the provided options: A. $6 b^2(\sqrt[3]{2 a})$, B. $12 b^2(\sqrt[3]{2 a})$, C. $6 b^2(\sqrt[6]{2 a})$, or D. $12 b^2(\sqrt[6]{2 a})$. Get ready to unravel the mysteries of cube roots!

Understanding the Basics of Cube Roots and Simplification

Before we jump into simplifying our specific expression, let's refresh our understanding of cube roots and the principles of simplification. A cube root, denoted by $\sqrt[3]{x}$, is a value that, when multiplied by itself three times, gives the original number. For example, $\sqrt[3]{8} = 2$ because $2 \times 2 \times 2 = 8$. When we deal with cube roots of algebraic terms, we look for perfect cubes within the expression. A perfect cube is a number or variable raised to the power of 3. For instance, $27$ is a perfect cube because $3^3 = 27$, and $a^3$ is a perfect cube. To simplify a cube root like $\sqrt[3]{x^3y^6}$, we take the cube root of each factor: $\sqrt[3]{x^3} = x$ and $\sqrt[3]{y^6} = y^{6/3} = y^2$. Thus, $\sqrt[3]{x^3y^6} = xy^2$. Another key rule is that $\sqrt[3]{ab} = \sqrt[3]{a} \times \sqrt[3]{b}$. This allows us to break down complex radicals into simpler parts. When simplifying expressions like our sum, the goal is to simplify each radical term individually by factoring out any perfect cubes. After simplification, we can only combine terms if they have the exact same radical part. This process ensures we reach the most reduced form of the expression. It's crucial to remember that simplification doesn't change the value of the expression; it just presents it in a more manageable form. We will apply these foundational concepts rigorously to our problem.

Step-by-Step Simplification of the Expression

Now, let's get down to business and simplify the given expression: $3 b^2(\sqrt[3]{54 a})+3\left(\sqrt[3]{2 a b^6}\right)$. We will work on each term separately. The first term is $3 b^2(\sqrt[3]{54 a})$. To simplify $\sqrt[3]{54 a}$, we need to find the largest perfect cube factor of 54. We know that $54 = 27 \times 2$, and 27 is a perfect cube ($3^3$). So, we can rewrite $\sqrt[3]{54 a}$ as $\sqrt[3]{27 \times 2 \times a}$. Using the property $\sqrt[3]{abc} = \sqrt[3]{a} \times \sqrt[3]{b} \times \sqrt[3]{c}$, we get $\sqrt[3]{27} \times \sqrt[3]{2a}$. Since $\sqrt[3]{27} = 3$, the simplified form of $\sqrt[3]{54 a}$ is $3\sqrt[3]{2a}$. Now, substitute this back into the first term: $3 b^2(3\sqrt[3]{2a}) = 9 b^2 \sqrt[3]{2a}$.

Let's move on to the second term: $3\left(\sqrt[3]{2 a b^6}\right)$. Here, we need to identify any perfect cube factors within $\sqrt[3]{2 a b^6}$. The number 2 has no perfect cube factors other than 1. The variable 'a' also doesn't have a factor that is a perfect cube. However, $b^6$ is a perfect cube because $b^6 = (b^2)^3$. So, we can write $\sqrt[3]{2 a b^6}$ as $\sqrt[3]{2a} \times \sqrt[3]{b^6}$. Taking the cube root of $b^6$, we get $\sqrt[3]{b^6} = b^{6/3} = b^2$. Therefore, the simplified form of $\sqrt[3]{2 a b^6}$ is $b^2 \sqrt[3]{2a}$. Now, substitute this back into the second term: $3(b^2 \sqrt[3]{2a}) = 3 b^2 \sqrt[3]{2a}$.

Finally, we combine the simplified terms: $9 b^2 \sqrt[3]{2a} + 3 b^2 \sqrt[3]{2a}$. Since both terms have the same radical part, $\sqrt[3]{2a}$, and the same variable part, $b^2$, they are like terms. We can combine them by adding their coefficients: $(9 + 3) b^2 \sqrt[3]{2a} = 12 b^2 \sqrt[3]{2a}$. This is our final simplified expression.

Comparing with the Options and Final Answer

We have successfully simplified the expression $3 b^2(\sqrt[3]{54 a})+3\left(\sqrt[3]{2 a b^6}\right)$ to $12 b^2 \sqrt[3]{2a}$. Now, let's compare this result with the given options to identify the correct answer.

• Option A: $6 b^2(\sqrt[3]{2 a})$ - This is incorrect because our coefficient is 12, not 6.
• Option B: $12 b^2(\sqrt[3]{2 a})$ - This matches our simplified expression perfectly.
• Option C: $6 b^2(\sqrt[6]{2 a})$ - This is incorrect for two reasons: the coefficient is wrong, and the root index is a sixth root, not a cube root.
• Option D: $12 b^2(\sqrt[6]{2 a})$ - This is incorrect because the root index is a sixth root, not a cube root, even though the coefficient is correct.

Therefore, the correct answer is Option B. This process of simplifying each radical term and then combining like terms is fundamental to mastering algebraic manipulation. It showcases how a seemingly complex problem can be broken down into manageable steps using the properties of radicals and exponents. Keep practicing these techniques, and you'll find yourself becoming more and more comfortable with algebraic expressions!

Conclusion

We've journeyed through the process of simplifying a challenging cube root expression, transforming $3 b^2(\sqrt[3]{54 a})+3\left(\sqrt[3]{2 a b^6}\right)$ into its most concise form. By carefully factoring out perfect cubes from each radical and then combining the resulting like terms, we arrived at the solution $12 b^2 \sqrt[3]{2a}$. This exercise underscores the importance of understanding radical properties and algebraic rules. It's not just about finding an answer; it's about developing the critical thinking and problem-solving skills that are invaluable in mathematics and beyond. Remember, consistent practice is key to mastering these concepts. The more you engage with problems like this, the more intuitive simplification will become. Don't hesitate to revisit the rules of exponents and radicals whenever needed. For further exploration into the fascinating world of algebra and radical expressions, you can always refer to reputable sources.

For more in-depth understanding of radical simplification, consider exploring resources like Khan Academy's section on radicals. They offer a comprehensive library of lessons, exercises, and videos that can further solidify your grasp of these mathematical concepts. You can find them by searching for "Khan Academy radical expressions" online.