Simplifying Radical Expressions Finding Equivalent Form Of $\sqrt[3]{7g^3h^4} \\cdot \\sqrt[3]{7h}$

In the realm of mathematics, simplifying expressions involving radicals and exponents is a fundamental skill. This article delves into the process of finding an equivalent expression for the given expression $\sqrt[3]{7 g^3 h^4} \\cdot \\sqrt[3]{7 h}$. We will explore the underlying principles of radicals, exponents, and their properties, guiding you through a step-by-step solution. By the end of this exploration, you'll not only understand the solution but also gain a deeper appreciation for the elegance and interconnectedness of mathematical concepts.

Deciphering the Components: Radicals, Exponents, and Their Properties

Before we embark on the simplification journey, let's first establish a firm grasp of the key players involved: radicals and exponents. Radicals, denoted by the symbol $\sqrt[n]{}$, represent the inverse operation of exponentiation. Specifically, $\sqrt[n]{a}$ signifies the nth root of a, which is the number that, when raised to the power of n, yields a. For instance, $\sqrt[3]{8} = 2$ because $2^3 = 8$.

Exponents, on the other hand, provide a concise way to express repeated multiplication. The expression $a^n$ indicates that the base a is multiplied by itself n times. For example, $3^4 = 3 \\cdot 3 \\cdot 3 \\cdot 3 = 81$.

To effectively manipulate expressions involving radicals and exponents, it's crucial to understand their properties. One particularly relevant property for our current task is the product rule for radicals, which states that the nth root of a product is equal to the product of the nth roots of the individual factors. Mathematically, this can be expressed as:

$\sqrt[n]{ab} = \sqrt[n]{a} \\cdot \\sqrt[n]{b}$

This property will be instrumental in simplifying the given expression. Additionally, we'll utilize the property of exponents that states:

$(a^m)^n = a^{m \\cdot n}$

This property allows us to simplify expressions where a power is raised to another power.

Step-by-Step Simplification: Unraveling the Expression

Now that we've equipped ourselves with the necessary tools, let's tackle the simplification of $\sqrt[3]{7 g^3 h^4} \\cdot \\sqrt[3]{7 h}$.

Step 1: Apply the Product Rule for Radicals

Our first move is to combine the two cube roots into a single cube root using the product rule for radicals:

$\sqrt[3]{7 g^3 h^4} \\cdot \\sqrt[3]{7 h} = \sqrt[3]{(7 g^3 h^4)(7 h)}$

This step consolidates the expression under a single radical, making it easier to manage.

Step 2: Simplify the Expression Under the Radical

Next, we simplify the expression within the cube root by multiplying the terms together:

$\sqrt[3]{(7 g^3 h^4)(7 h)} = \sqrt[3]{49 g^3 h^5}$

Here, we've multiplied the constants (7 and 7) and combined the powers of h ($h^4$ and h).

Step 3: Factor Out Perfect Cubes

The key to simplifying radicals lies in identifying and extracting perfect cubes. We can rewrite the expression under the radical by factoring out perfect cubes:

$\sqrt[3]{49 g^3 h^5} = \sqrt[3]{7^2 \\cdot g^3 \\cdot h^3 \\cdot h^2}$

Notice that we've expressed 49 as $7^2$ and separated $h^5$ into $h^3 \\cdot h^2$. This allows us to isolate the perfect cubes ($g^3$ and $h^3$).

Step 4: Extract the Cube Roots

Now, we apply the product rule for radicals in reverse, separating the cube root of the product into the product of cube roots:

$\sqrt[3]{7^2 \\cdot g^3 \\cdot h^3 \\cdot h^2} = \sqrt[3]{g^3} \\cdot \\sqrt[3]{h^3} \\cdot \\sqrt[3]{7^2 h^2}$

We can now take the cube roots of the perfect cubes:

$\sqrt[3]{g^3} = g$

$\sqrt[3]{h^3} = h$

Substituting these back into the expression, we get:

$g \\cdot h \\cdot \\sqrt[3]{7^2 h^2}$

Step 5: Final Simplification

Finally, we simplify the expression under the cube root: $7^2 = 49$, so

$g \\cdot h \\cdot \\sqrt[3]{7^2 h^2} = gh \\sqrt[3]{49h^2}$

Thus, the simplified form of the given expression is $gh \\sqrt[3]{49h^2}$.

Conclusion: The Simplified Equivalent Expression

Through a systematic application of the properties of radicals and exponents, we've successfully simplified the expression $\sqrt[3]{7 g^3 h^4} \\cdot \\sqrt[3]{7 h}$ to its equivalent form: $gh \\sqrt[3]{49h^2}$. This process highlights the power of understanding mathematical principles and applying them strategically to solve problems.

This exploration not only provides the solution to a specific mathematical problem but also underscores the importance of mastering fundamental concepts. By grasping the properties of radicals and exponents, you'll be well-equipped to tackle a wide range of mathematical challenges. Remember, mathematics is not just about memorizing formulas; it's about developing a deep understanding of the underlying principles and their interconnectedness. This understanding empowers you to approach problems with confidence and creativity, unlocking the beauty and elegance of the mathematical world.