Solving $5^{\frac{1}{4}} imes 5^{\frac{4}{5}}$: A Step-by-Step Guide

Understanding exponents and how they interact with each other is crucial in mathematics. This article will guide you through the process of solving the expression $5^{\frac{1}{4}} imes 5^{\frac{4}{5}}$, breaking down each step to ensure clarity. By the end of this guide, you'll not only have the solution but also a deeper understanding of the underlying principles.

Understanding the Basics of Exponents

Before diving into the problem, let's establish a solid understanding of exponents. Exponents, also known as powers, indicate how many times a number (the base) is multiplied by itself. For instance, in the expression $a^b$, 'a' is the base, and 'b' is the exponent. The exponent tells us how many times to multiply the base by itself.

When dealing with fractional exponents, such as in our problem, it's important to recognize their connection to roots. A fractional exponent like $\frac{1}{n}$ indicates the nth root of the base. For example, $x^{\frac{1}{2}}$ is the square root of x, and $x^{\frac{1}{3}}$ is the cube root of x. Understanding this relationship is key to simplifying expressions with fractional exponents.

Another fundamental rule is the product of powers rule. This rule states that when multiplying two powers with the same base, you can add their exponents. Mathematically, this is expressed as $a^m imes a^n = a^{m+n}$. This rule is essential for solving our problem, as it allows us to combine the two terms into a single expression with a single exponent. Keep this rule in mind as we proceed through the solution.

Step 1: Applying the Product of Powers Rule

The first step in solving the expression $5^{\frac{1}{4}} imes 5^{\frac{4}{5}}$ is to apply the product of powers rule. As mentioned earlier, this rule states that when multiplying powers with the same base, we add the exponents. In this case, our base is 5, and our exponents are $\frac{1}{4}$ and $\frac{4}{5}$. Therefore, we can rewrite the expression as:

$5^{\frac{1}{4} + \frac{4}{5}}$

This step simplifies the problem by combining the two terms into a single power of 5. Now, our focus shifts to adding the fractions in the exponent. This requires finding a common denominator, which we will address in the next step. Remember, the key to mastering exponents is to break down complex problems into simpler steps, and the product of powers rule is a powerful tool in this process.

Step 2: Adding the Exponents

Now that we have the expression $5^{\frac{1}{4} + \frac{4}{5}}$, the next step involves adding the fractions in the exponent. To add fractions, they must have a common denominator. In this case, the denominators are 4 and 5. The least common multiple (LCM) of 4 and 5 is 20. Therefore, we need to convert both fractions to have a denominator of 20.

To convert $\frac{1}{4}$ to a fraction with a denominator of 20, we multiply both the numerator and the denominator by 5:

$\frac{1}{4} imes \frac{5}{5} = \frac{5}{20}$

Similarly, to convert $\frac{4}{5}$ to a fraction with a denominator of 20, we multiply both the numerator and the denominator by 4:

$\frac{4}{5} imes \frac{4}{4} = \frac{16}{20}$

Now that both fractions have the same denominator, we can add them:

$\frac{5}{20} + \frac{16}{20} = \frac{21}{20}$

So, the sum of the exponents is $\frac{21}{20}$. This means our expression now becomes:

$5^{\frac{21}{20}}$

We have successfully added the exponents, which brings us closer to the final solution. In the next step, we will explore how to interpret and potentially simplify this result.

Step 3: Interpreting and Simplifying the Result

At this point, we have simplified the expression to $5^{\frac{21}{20}}$. This is a valid answer, but we can further interpret it to gain a better understanding. The fractional exponent $\frac{21}{20}$ can be seen as a combination of a whole number and a fraction. Specifically, $\frac{21}{20}$ is equal to $1 + \frac{1}{20}$. This allows us to rewrite the expression using the product of powers rule in reverse.

Recall that $a^{m+n} = a^m imes a^n$. Applying this rule, we can rewrite $5^{\frac{21}{20}}$ as:

$5^{1 + \frac{1}{20}} = 5^1 imes 5^{\frac{1}{20}}$

This simplifies to:

$5 imes 5^{\frac{1}{20}}$

Now, let's interpret $5^{\frac{1}{20}}$. As we discussed earlier, a fractional exponent of the form $\frac{1}{n}$ represents the nth root. Therefore, $5^{\frac{1}{20}}$ is the 20th root of 5, which can be written as $\sqrt[20]{5}$. So, our expression becomes:

$5 imes \sqrt[20]{5}$

This is the simplified form of the expression. It's important to note that while we have simplified the expression, the numerical value remains the same. We have simply expressed it in a more understandable form.

Step 4: Approximating the Numerical Value (Optional)

While $5 imes \sqrt[20]{5}$ is the simplified form of the expression, we can also approximate its numerical value using a calculator. This step is optional but can provide a better sense of the magnitude of the result.

Using a calculator, we find that the 20th root of 5 is approximately 1.0838. Therefore, the expression becomes:

$5 imes 1.0838 \approx 5.419$

So, the approximate numerical value of $5^{\frac{1}{4}} imes 5^{\frac{4}{5}}$ is 5.419. This gives us a concrete value for the expression, making it easier to compare with other numbers.

Conclusion

In this article, we have successfully solved the expression $5^{\frac{1}{4}} imes 5^{\frac{4}{5}}$ by applying the rules of exponents. We started by understanding the basics of exponents, including fractional exponents and the product of powers rule. Then, we systematically worked through the problem, adding the exponents, interpreting the result, and simplifying it into a more understandable form. Finally, we approximated the numerical value to provide a concrete understanding of the result.

By breaking down the problem into manageable steps, we were able to navigate the complexities of exponents and arrive at the solution. This approach is crucial for tackling more challenging mathematical problems. Remember to always focus on understanding the underlying principles and applying them methodically. With practice, you'll become more confident and proficient in solving exponential expressions.

For further exploration of exponential functions and their properties, you might find the resources at Khan Academy's Algebra section helpful.