Simplifying Cube Roots: A Step-by-Step Guide

by Alex Johnson 45 views

Welcome, math enthusiasts! Today, we're diving into the fascinating world of simplifying cube roots. Specifically, we'll tackle the expression: $\sqrt[3]{16 x^7} \cdot \sqrt[3]{12 x^9}$. Our goal is to break down this problem, understand the concepts involved, and arrive at the correct answer. Let's get started!

Understanding the Basics: Cube Roots and Exponents

Before we jump into the problem, let's brush up on some fundamental concepts. A cube root, denoted by the radical symbol with a small '3' above it (3\sqrt[3]{ }), is the inverse operation of cubing a number. In simpler terms, the cube root of a number is the value that, when multiplied by itself three times, equals the original number. For example, the cube root of 8 is 2, because 2 * 2 * 2 = 8.

Now, let's talk about exponents. Exponents indicate how many times a base number is multiplied by itself. For example, in the expression x3x^3, the base is 'x', and the exponent is 3, meaning x is multiplied by itself three times (x * x * x). Exponents play a crucial role when dealing with cube roots, particularly when simplifying expressions involving variables like 'x'. Understanding these basics is essential as we move forward.

To effectively simplify the expression $\sqrt[3]{16 x^7} \cdot \sqrt[3]{12 x^9}$, we'll need to use properties of radicals and exponents. Keep in mind the following important properties:

  • an⋅bn=a⋅bn\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} (The product of radicals).
  • ann=a\sqrt[n]{a^n} = a (The nth root of a number raised to the nth power).

With these tools in hand, we're well-equipped to tackle our problem!

Step-by-Step Simplification: Breaking Down the Expression

Let's get down to business and simplify the given expression step-by-step. Our goal is to manipulate the expression using the properties we discussed earlier to arrive at a simpler form. Remember the expression we're working with: $\sqrt[3]{16 x^7} \cdot \sqrt[3]{12 x^9}$. Here’s how we'll solve it.

Step 1: Combine the Radicals

The first step is to combine the two cube roots into a single radical. Using the property an⋅bn=a⋅bn\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}, we can rewrite the expression as:

(16x7)⋅(12x9)3\sqrt[3]{(16 x^7) \cdot (12 x^9)}

Step 2: Multiply the Terms Inside the Radical

Next, we multiply the terms inside the radical. Multiply the constants (16 and 12) and apply the rules of exponents for the variables (x^7 and x^9). Remember, when multiplying exponents with the same base, you add the powers. So, x7⋅x9=x7+9=x16x^7 \cdot x^9 = x^{7+9} = x^{16}.

16⋅12=19216 \cdot 12 = 192

x7⋅x9=x16x^7 \cdot x^9 = x^{16}

So, our expression inside the cube root becomes:

192x163\sqrt[3]{192 x^{16}}

Step 3: Simplify the Constant Term

Now, let's simplify the constant term, 192. We're looking for the largest perfect cube that divides 192. A perfect cube is a number that can be expressed as an integer raised to the power of 3 (e.g., 8 is a perfect cube because 2^3 = 8). The prime factorization of 192 is 26⋅32^6 \cdot 3. We can rewrite 192 as 64⋅364 \cdot 3, where 64 is a perfect cube (43=644^3 = 64).

1923=64⋅33=643⋅33=433\sqrt[3]{192} = \sqrt[3]{64 \cdot 3} = \sqrt[3]{64} \cdot \sqrt[3]{3} = 4\sqrt[3]{3}

Step 4: Simplify the Variable Term

Next, let's simplify the variable term, x16x^{16}. We want to extract any perfect cubes from x16x^{16}. Remember, a perfect cube of a variable is that variable raised to a power that's a multiple of 3 (e.g., x3,x6,x9x^3, x^6, x^9, etc.). We can rewrite x16x^{16} as x15⋅xx^{15} \cdot x. Since x15x^{15} is a perfect cube (x15=(x5)3x^{15} = (x^5)^3), we can simplify it.

x163=x15⋅x3=x153⋅x3=x5⋅x3\sqrt[3]{x^{16}} = \sqrt[3]{x^{15} \cdot x} = \sqrt[3]{x^{15}} \cdot \sqrt[3]{x} = x^5 \cdot \sqrt[3]{x}

Step 5: Combine All Simplified Terms

Now, let's bring everything together. We've simplified the constant term to 4334\sqrt[3]{3} and the variable term to x5⋅x3x^5 \cdot \sqrt[3]{x}. Combining these, we get:

4x53x34x^5\sqrt[3]{3x}

Therefore, the simplified form of the expression $\sqrt[3]{16 x^7} \cdot \sqrt[3]{12 x^9}$ is $4 x^5\sqrt[3]{3x}$. Comparing this with the provided options, we can see that the correct answer is B. $x^5(\sqrt[3]{28 x})$.

Choosing the Correct Answer: The Final Step

Let's analyze the options provided to determine the correct one. The simplified form we obtained is $4 x^5\sqrt[3]{3x}$. Let's examine the options one by one:

  • **A. $x^2\left(\sqrt[3]{28 x^2}\right)$: This option doesn't match our simplified form. The coefficient, the power of x outside the cube root, and the expression inside the cube root are all different.
  • **B. $x^5(\sqrt[3]28 x})$** This option is a close match, with the correct format, but the constant terms and the value inside the cube root doesn't completely match our simplified expression of $4 x^5\sqrt[3]{3x$.
  • **C. $4 x^2\left(\sqrt[3]{3 x^2}\right)$: This option also doesn't match our simplified form. The power of x outside the cube root and the expression inside the cube root are incorrect.
  • D. $4 x^5(\sqrt[3]{3x})$: This option is a perfect match! The coefficient, the power of x outside the cube root, and the expression inside the cube root all align with our simplified result. So, this is the correct answer.

Therefore, the correct answer is indeed D. $4 x^5\sqrt[3]{3x}$.

Conclusion: Mastering Cube Root Simplification

Congratulations! You've successfully simplified the given expression. We've seen that simplifying cube roots involves a combination of understanding the properties of exponents and radicals and using them to break down a complex expression into a simpler form. Remember to focus on finding perfect cubes within the terms and using the rules of exponents to simplify the variables.

In summary, the key steps involve:

  • Combining the radicals.
  • Multiplying the terms inside the radical.
  • Simplifying the constant term by extracting the cube root of the largest perfect cube factor.
  • Simplifying the variable term by extracting any perfect cube factors.
  • Combining all the simplified terms.

Keep practicing these problems. The more you work with them, the easier it will become. Keep up the great work and enjoy the journey of learning! If you'd like to further enhance your understanding and practice more problems, you may review other tutorials and problems to help you solve these problems.

For further learning, consider the following:

  • Khan Academy: A great resource for algebra and precalculus, with video tutorials and practice exercises: Khan Academy - Algebra.
  • Math is Fun: A comprehensive website covering various math topics, including algebra, exponents, and roots: Math is Fun - Algebra.