Simplify (48^(1/4) / 6^(1/4))^6: Step-by-Step Solution

Mathematics can sometimes feel like a complex puzzle, and simplifying expressions is a key skill to mastering it. Today, we're going to tackle a problem that involves fractional exponents and roots: simplify $\left(\frac{48^{1/4}}{6^{1/4}}\right)^6$. This might look a bit intimidating at first glance, but by breaking it down step by step, we can easily find the solution. We'll explore the properties of exponents that make this simplification possible, making sure to explain each part clearly so you can confidently approach similar problems in the future. Get ready to dive into the elegant world of mathematical simplification!

Understanding the Exponent Rules

Before we start simplifying the expression $\left(\frac{48^{1/4}}{6^{1/4}}\right)^6$, it's crucial to understand a few fundamental rules of exponents. These rules are the building blocks that allow us to manipulate and simplify expressions involving powers. The first rule we'll use is the quotient of powers with the same exponent: $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$. This rule states that if you are dividing two numbers that have the same exponent, you can divide the bases first and then apply the common exponent. This is incredibly useful because it allows us to combine terms that look separate into a single, more manageable expression. Another key rule is the power of a power: $(a^m)^n = a^{m \times n}$. This rule tells us that if you raise a power to another power, you multiply the exponents. These two rules, when applied correctly, will unlock the solution to our problem. It's like having a secret code to unlock the complexity of the expression. By internalizing these rules, you gain the power to simplify a wide range of algebraic expressions, making your journey through mathematics much smoother and more enjoyable. Remember, practice is key to mastering these rules, so don't hesitate to try them out on different problems.

Step 1: Applying the Quotient of Powers Rule

Our expression is $\left(\frac{48^{1/4}}{6^{1/4}}\right)^6$. Notice that both the numerator and the denominator have the same exponent, which is $1/4$. This is the perfect scenario to apply the quotient of powers rule we just discussed: $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$. In our case, $a=48$, $b=6$, and $n=1/4$. So, we can rewrite the expression inside the parentheses as follows: $\frac{48^{1/4}}{6^{1/4}} = \left(\frac{48}{6}\right)^{1/4}$. Now, let's simplify the fraction inside the parentheses: $\frac{48}{6} = 8$. So, the expression inside the parentheses becomes $8^{1/4}$. This step significantly reduces the complexity of our problem, transforming a fraction of powers into a single base raised to a fractional power. It's a satisfying transformation that shows the power of applying the right mathematical rules. This simplification is a crucial stepping stone towards the final answer, and it demonstrates how seemingly complicated expressions can be elegantly unraveled by understanding fundamental algebraic principles. Keep this simplified form in mind as we move to the next step.

Step 2: Applying the Power of a Power Rule

After the first step, our expression has been simplified to $\left(8^{1/4}\right)^6$. Now, we need to deal with the outer exponent, which is $6$. This is where the power of a power rule comes into play: $(a^m)^n = a^{m \times n}$. In our current expression, $a=8$, $m=1/4$, and $n=6$. According to the rule, we need to multiply the exponents: $m \times n = \frac{1}{4} \times 6$. Calculating this product gives us $\frac{6}{4}$. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is $2$. So, $\frac{6}{4} = \frac{3}{2}$. Therefore, our expression becomes $8^{3/2}$. This is a major simplification, bringing us very close to the final answer. We've successfully combined the two exponents using a fundamental property of powers, resulting in a much simpler form that is easier to evaluate. This step highlights how exponents can be combined and manipulated to reduce the complexity of mathematical expressions.

Step 3: Evaluating the Final Expression

We have now arrived at the simplified form of our original expression, which is $8^{3/2}$. The final step is to evaluate this expression. Remember that an exponent of the form $\frac{p}{q}$ means taking the $q$-th root and then raising it to the power of $p$. In our case, the exponent is $3/2$, so $p=3$ and $q=2$. This means we need to take the square root of $8$ and then cube the result. Mathematically, this is written as $(\sqrt{8})^3$. Let's first find the square root of $8$. We can write $8$ as $4 \times 2$. So, $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$. Now, we need to cube this result: $(2\sqrt{2})^3$. This means multiplying $2\sqrt{2}$ by itself three times: $(2\sqrt{2}) \times (2\sqrt{2}) \times (2\sqrt{2})$. Let's multiply the numerical parts first: $2 \times 2 \times 2 = 8$. Now, let's multiply the radical parts: $\sqrt{2} \times \sqrt{2} \times \sqrt{2} = (\sqrt{2} \times \sqrt{2}) \times \sqrt{2} = 2 \times \sqrt{2} = 2\sqrt{2}$. Combining the numerical and radical parts, we get $8 \times 2\sqrt{2} = 16\sqrt{2}$.

Alternatively, we can evaluate $8^{3/2}$ by first cubing $8$ and then taking the square root. $8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512$. Then, we take the square root of $512$: $\sqrt{512}$. We can simplify $\sqrt{512}$ by finding the largest perfect square factor of $512$. $512 = 256 \times 2$, and $256$ is a perfect square ($16^2$). So, $\sqrt{512} = \sqrt{256 \times 2} = \sqrt{256} \times \sqrt{2} = 16\sqrt{2}$. Both methods lead to the same result.

Final Answer Check

Let's quickly review the steps to ensure accuracy. We started with $\left(\frac{48^{1/4}}{6^{1/4}}\right)^6$. We applied the quotient rule to get $\left(8^{1/4}\right)^6$. Then, we applied the power of a power rule to get $8^{3/2}$. Finally, we evaluated $8^{3/2}$ as $16\sqrt{2}$. Let's double-check the provided options: $8^{1/4}$, $8^{3/2}$, $12$, and $262,144$. Our result, $16\sqrt{2}$, is not directly listed. However, it's important to re-examine the question and the options provided. Sometimes, simplification might lead to a form that isn't immediately obvious among the choices. Let's re-evaluate $8^{3/2}$ carefully. We know $8 = 2^3$. So, $8^{3/2} = (2^3)^{3/2}$. Using the power of a power rule again, we multiply the exponents: $3 \times \frac{3}{2} = \frac{9}{2}$. So, $8^{3/2} = 2^{9/2}$. This doesn't seem to simplify to any of the integer options easily. Let's go back to $8^{3/2} = (\sqrt{8})^3 = (2\sqrt{2})^3 = 16\sqrt{2}$.

Let's consider if there was a misunderstanding of the question or the options. It's possible that the expected answer is in a different form, or there's a mistake in the provided options or the original problem. However, based on standard mathematical rules, $16\sqrt{2}$ is the correct simplification of $8^{3/2}$.

Let's revisit the intermediate step $8^{3/2}$. If we interpret $8^{3/2}$ as $(8^3)^{1/2}$, we get $(512)^{1/2}$. If we interpret it as $(8^{1/2})^3$, we get $(\sqrt{8})^3 = (2\sqrt{2})^3 = 16\sqrt{2}$.

Let's reconsider the original expression and the possibility of integer answers. The presence of integer options like '12' and '262,144' suggests that perhaps the expression simplifies to an integer. Let's re-examine the exponent rules and calculations.

$\left(\frac{48^{1/4}}{6^{1/4}}\right)^6 = \left(\left(\frac{48}{6}\right)^{1/4}\right)^6 = \left(8^{1/4}\right)^6 = 8^{(1/4) \times 6} = 8^{6/4} = 8^{3/2}$.

Now, let's try to evaluate $8^{3/2}$ by thinking about prime factorization. $8 = 2^3$. So, $8^{3/2} = (2^3)^{3/2} = 2^{3 \times 3/2} = 2^{9/2}$. This is $2^{4.5}$.

Let's review the options again: $8^{1/4}$, $8^{3/2}$, $12$, $262,144$. The option $8^{3/2}$ is indeed one of the intermediate steps and also the simplified form of the expression. This is a strong candidate if the question is asking for the simplified form before full evaluation into a decimal or radical.

However, if we are expected to evaluate it further, let's assume there might be a typo in the question or options, as $16\sqrt{2}$ is not among the options. Let's check if any of the integer options can be reached through a plausible misinterpretation or a different path.

Consider the option '12'. It's hard to see how $8^{3/2}$ could become 12.

Consider the option '262,144'. This is $2^{18}$. Our result is $2^{9/2}$. There's no direct relation.

Let's go back to the most direct simplification: $\left(\frac{48^{1/4}}{6^{1/4}}\right)^6 = 8^{3/2}$. Given the options, it is highly probable that the question intends for the answer to be the expression $8^{3/2}$ itself, or there is a mistake in the question or the provided options. If we are forced to choose from the given options and interpret 'simplify' as reaching the most reduced exponential form, then $8^{3/2}$ is the most appropriate answer among the choices provided, as it is the direct result of applying exponent rules.

Let's consider if there's another way to interpret the roots. $48^{1/4}$ is the fourth root of 48. $6^{1/4}$ is the fourth root of 6. $\frac{\sqrt[4]{48}}{\sqrt[4]{6}} = \sqrt[4]{\frac{48}{6}} = \sqrt[4]{8}$. Then $(\sqrt[4]{8})^6 = (8^{1/4})^6 = 8^{6/4} = 8^{3/2}$. This confirms our calculation.

Since $8^{3/2}$ is an option, and it's the direct result of simplifying the expression using exponent rules, this is the most logical answer. The evaluation to $16\sqrt{2}$ is correct, but if $16\sqrt{2}$ is not an option, then the question likely expects the answer in the form $8^{3/2}$.

Conclusion

We have successfully simplified the expression $\left(\frac{48^{1/4}}{6^{1/4}}\right)^6$ by carefully applying the rules of exponents. The process involved using the quotient of powers rule and the power of a power rule, leading us to the intermediate simplified form of $8^{3/2}$. While further evaluation can result in $16\sqrt{2}$, given the provided options, $8^{3/2}$ (8 raised to the 3 halves power) is the most fitting answer that represents the simplified exponential form of the original expression. This exercise highlights the importance of understanding and applying exponent properties to make complex mathematical expressions manageable and to arrive at accurate solutions. Remember that practice is key to mastering these concepts. For further exploration of exponent rules and algebraic simplification, you can visit Khan Academy's Algebra section.