Simplifying Radicals: Solving $\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}$

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Have you ever stumbled upon a mathematical expression that looks intimidating at first glance but turns out to be surprisingly simple? The expression $\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}$ is a perfect example of this. In this article, we will break down this radical expression step by step, making it easy to understand and solve. Whether you are a student brushing up on your math skills or simply a curious mind eager to learn, this guide will provide you with a clear and concise explanation.

Understanding Radicals

Before we dive into the specifics of our expression, let's take a moment to understand what radicals are. Radicals, in mathematics, are used to represent roots of numbers. The most common radical is the square root, denoted by the symbol ${\sqrt{\ }}$. The square root of a number x is a value that, when multiplied by itself, gives x. For instance, the square root of 9 (written as ${\sqrt{9}}$) is 3, because 3 * 3 = 9.

But radicals aren't limited to square roots. We also have cube roots, fourth roots, fifth roots, and so on. The nth root of a number x is a value that, when raised to the power of n, gives x. This is denoted by ${\sqrt[n]{x}}$, where n is the index of the radical. In our expression, we are dealing with a fourth root, indicated by the index 4 in $\sqrt[4]{7}$.

Breaking Down the Fourth Root

In our problem, we encounter the fourth root of 7, written as $\sqrt[4]{7}$. The fourth root of a number is a value that, when multiplied by itself four times, equals that number. So, $\sqrt[4]{7}$ is a number that, when raised to the power of 4, gives us 7. While we can't express $\sqrt[4]{7}$ as a simple integer or fraction, we can work with it using the properties of radicals.

Understanding the concept of radicals is crucial for simplifying expressions like $\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}$. Now that we have a solid grasp of what radicals represent, let's move on to simplifying the given expression.

Simplifying the Expression: Step-by-Step

Now, let's tackle the expression $\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}$. This might look daunting, but we can simplify it using a fundamental property of radicals: when multiplying radicals with the same index, we can multiply the radicands (the numbers inside the radical) while keeping the index the same. In mathematical terms:

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

This property is our key to simplifying the expression.

Step 1: Grouping the Radicals

Our expression consists of four instances of $\sqrt[4]{7}$ multiplied together. Let's group them in pairs to make the simplification process clearer:

$(\sqrt[4]{7} \cdot \sqrt[4]{7}) \cdot (\sqrt[4]{7} \cdot \sqrt[4]{7})$

Step 2: Applying the Multiplication Rule

Now, we apply the multiplication rule to each pair of radicals. For the first pair, $\sqrt[4]{7} \cdot \sqrt[4]{7}$, we multiply the radicands (7 and 7) while keeping the index (4) the same:

$\sqrt[4]{7} \cdot \sqrt[4]{7} = \sqrt[4]{7 \cdot 7} = \sqrt[4]{49}$

Similarly, for the second pair:

$\sqrt[4]{7} \cdot \sqrt[4]{7} = \sqrt[4]{7 \cdot 7} = \sqrt[4]{49}$

Our expression now looks like this:

$\sqrt[4]{49} \cdot \sqrt[4]{49}$

Step 3: Multiplying Again

We still have two radicals multiplied together, so let's apply the multiplication rule once more:

$\sqrt[4]{49} \cdot \sqrt[4]{49} = \sqrt[4]{49 \cdot 49}$

Now we need to calculate 49 * 49. You can do this manually or use a calculator. 49 * 49 = 2401. So our expression becomes:

$\sqrt[4]{2401}$

Step 4: Simplifying the Result

We're getting closer! Now we have $\sqrt[4]{2401}$. To simplify this, we need to find a number that, when raised to the power of 4, equals 2401. In other words, we're looking for the fourth root of 2401.

You might recognize 2401 as a power of 7. In fact, 2401 = 7 * 7 * 7 * 7, which can be written as $7^4$. Therefore:

$\sqrt[4]{2401} = \sqrt[4]{7^4}$

The fourth root of $7^4$ is simply 7. So, our final simplified answer is:

$\sqrt[4]{2401} = 7$

Alternative Approach: Using Exponents

There's another way to simplify the expression $\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}$ that involves using exponents. This approach can be more intuitive for some, as it directly applies the rules of exponents.

Step 1: Convert Radicals to Exponents

The first step is to convert the radicals into exponential form. Recall that the nth root of a number x can be written as x raised to the power of 1/n. In other words:

$\sqrt[n]{x} = x^{\frac{1}{n}}$

Applying this to our expression, we can rewrite $\sqrt[4]{7}$ as $7^{\frac{1}{4}}$. So, our expression becomes:

$7^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot 7^{\frac{1}{4}}$

Step 2: Apply the Product of Powers Rule

Now we have a product of powers with the same base (7). The product of powers rule states that when multiplying powers with the same base, you add the exponents:

$a^m \cdot a^n = a^{m+n}$

Applying this rule to our expression, we add the exponents:

$7^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} = 7^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}}$

Step 3: Simplify the Exponent

Now, we need to simplify the exponent. Adding the fractions, we get:

$\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{4}{4} = 1$

So our expression becomes:

$7^1$

Step 4: Final Result

Any number raised to the power of 1 is simply the number itself. Therefore:

$7^1 = 7$

As you can see, this approach leads us to the same answer as before: 7. Both methods, simplifying radicals directly and using exponents, are valid and useful. The choice of which method to use often comes down to personal preference and the specific problem at hand.

Conclusion

In this article, we've successfully simplified the expression $\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}$. We explored two methods: directly simplifying the radicals and converting them to exponents. Both approaches demonstrate how seemingly complex expressions can be broken down into manageable steps using fundamental mathematical principles.

Whether you're studying for a math exam or simply enjoy the challenge of problem-solving, understanding radicals and exponents is a valuable skill. By practicing these techniques, you'll become more confident in your ability to tackle a wide range of mathematical problems.

For further exploration of radical simplification and exponent rules, you may find helpful resources on websites like Khan Academy's Algebra I section.