Simplifying The Expression: $2^{1/4} + 1^{1/3} + 5/6

Hey guys! Let's break down this mathematical expression together. It might look a bit intimidating at first, but we'll take it step by step to make sure we understand each part. Our main goal here is to simplify $2^{1/4} + 1^{1/3} + \frac{5}{6}$ and figure out which of the provided options is the correct one. Stick with me, and you'll see it's totally manageable! We'll use some basic math principles and a bit of algebraic thinking to solve this. So, let's dive in and get started!

Understanding the Components

Okay, first things first, let's look at each part of the expression individually. Understanding each component is crucial before we try to simplify the whole thing. This way, we avoid getting lost in the complexity and can tackle each term effectively. Remember, breaking down a problem into smaller, more manageable parts is a key strategy in math (and in life, honestly!).

The Term $2^{1/4}$

When you see an expression like $2^{1/4}$, it's important to recognize that this is another way of writing a radical. Specifically, $2^{1/4}$ is the same as the fourth root of 2, often written as $\sqrt[4]{2}$. The denominator in the fractional exponent tells you the type of root you're dealing with. So, in this case, we're looking for a number that, when multiplied by itself four times, gives you 2. Now, the fourth root of 2 isn't a whole number; it's an irrational number. This means it has a decimal representation that goes on forever without repeating. To give you a sense of its value, $\sqrt[4]{2}$ is approximately 1.189. But for our purposes right now, we'll keep it in its exact form, $2^{1/4}$, because trying to work with the decimal approximation would make the problem much more complicated. We'll see how this term fits into the bigger picture as we move forward. The key thing to remember is that fractional exponents represent roots, and understanding this relationship is crucial for simplifying expressions like this.

The Term $1^{1/3}$

Next up, we have $1^{1/3}$. This might seem a bit trickier at first glance, but it's actually quite simple. Remember, any number raised to the power of $\frac{1}{3}$ is the same as taking the cube root of that number. So, $1^{1/3}$ is the same as $\sqrt[3]{1}$, which is the cube root of 1. Now, what number multiplied by itself three times equals 1? Well, that's easy – it's just 1! So, $1^{1/3} = 1$. This is a fantastic simplification because it turns what looks like a complicated term into a simple whole number. This makes our overall expression much easier to handle. When you encounter terms like this, always remember the fundamental definitions of exponents and roots; they can often lead to quick and straightforward simplifications. It's one of those little math secrets that makes a big difference!

The Fraction $\frac{5}{6}$

Our last piece of the puzzle is the fraction $\frac{5}{6}$. This one is already in a pretty simple form, but it's important to understand how it fits into the overall expression. Fractions represent parts of a whole, and in this case, we have five-sixths of something. To work with this fraction alongside the other terms, we'll likely need to find a common denominator when we add things together. This is a standard procedure when dealing with fractions, and it ensures we're comparing and combining like quantities. For now, we'll just keep $\frac{5}{6}$ as it is, but we'll keep in mind that we'll need to handle it carefully when we combine it with the other terms. Fractions might seem basic, but they're a fundamental part of math, and mastering them is crucial for more complex problems. So, we've got our fraction ready and waiting for the next step!

Combining the Simplified Terms

Alright, now that we've simplified each term individually, let's bring them all together and see what we've got. We started with the expression $2^{1/4} + 1^{1/3} + \frac{5}{6}$. We figured out that $2^{1/4}$ is the fourth root of 2, which we'll keep in that form for now. We also simplified $1^{1/3}$ to just 1. And we have the fraction $\frac{5}{6}$ as it is. So, our expression now looks like this: $2^{1/4} + 1 + \frac{5}{6}$. It's looking a bit cleaner already, isn't it? Now, the next step is to combine these terms into a single, simplified expression. This means we'll need to add the whole number and the fraction together, and then see how the fourth root of 2 fits in. This is where our algebraic skills come in handy. Get ready, because we're about to put all the pieces together!

Adding the Whole Number and the Fraction

Okay, let's tackle the easiest part first: adding the whole number 1 and the fraction $\frac{5}{6}$. To do this, we need to express 1 as a fraction with the same denominator as $\frac{5}{6}$. Remember, any whole number can be written as a fraction by putting it over 1. So, 1 is the same as $\frac{1}{1}$. Now, to get a denominator of 6, we multiply both the numerator and the denominator of $\frac{1}{1}$ by 6. This gives us $\frac{6}{6}$. So, now we have $1 = \frac{6}{6}$. Now we can easily add $\frac{6}{6}$ and $\frac{5}{6}$. When you add fractions with the same denominator, you just add the numerators and keep the denominator the same. So, $\frac{6}{6} + \frac{5}{6} = \frac{6 + 5}{6} = \frac{11}{6}$. Great! We've combined the 1 and the $\frac{5}{6}$ into a single fraction, $\frac{11}{6}$. This makes our expression even simpler. We're on a roll here! Now, let's see how this fits in with the remaining term, $2^{1/4}$.

Putting It All Together

Now, let's bring everything together. We've simplified our original expression to $2^{1/4} + 1 + \frac{5}{6}$. We then combined the 1 and the $\frac{5}{6}$ to get $\frac{11}{6}$. So, our expression now looks like this: $2^{1/4} + \frac{11}{6}$. This is as simplified as we can get without using a calculator to approximate the value of $2^{1/4}$. The term $2^{1/4}$ is an irrational number, and we can't simplify it further into a simple fraction or whole number. So, we're left with the sum of the fourth root of 2 and the fraction $\frac{11}{6}$. Now, let's take a look at our options and see which one matches our simplified expression.

Checking the Options

Alright, we've done the hard work of simplifying the expression. Now comes the moment of truth: matching our simplified expression with one of the options provided. Our simplified expression is $2^{1/4} + \frac{11}{6}$. Let's quickly recap the options we have:

(a) $\frac{3^7}{12}$ (b) $\frac{4^{5}}{12}$ (c) $\frac{6}{12}$ (d) $\frac{85}{12}$ (e) $\frac{8^7}{12}$

To find the correct answer, we need to see if any of these options are equivalent to $2^{1/4} + \frac{11}{6}$. This might involve some clever manipulation or approximation. Let's go through each option one by one and see if we can make a match. Remember, we're looking for the option that, when simplified, gives us something that looks like $2^{1/4} + \frac{11}{6}$. So, let's put on our detective hats and start investigating!

Evaluating Option (d) $\frac{85}{12}$

Let's start by examining option (d), which is $\frac{85}{12}$. This looks like a promising candidate because it's a single fraction, and we have a fraction in our simplified expression. To see if it matches, we need to compare it to $2^{1/4} + \frac{11}{6}$. The first thing we can do is try to express $\frac{11}{6}$ with a denominator of 12, so we can compare the fractional parts more easily. To do this, we multiply both the numerator and the denominator of $\frac{11}{6}$ by 2, which gives us $\frac{22}{12}$. So, now we're comparing $2^{1/4} + \frac{22}{12}$ to $\frac{85}{12}$. If option (d) is correct, then $2^{1/4}$ must be equal to $\frac{85}{12} - \frac{22}{12}$. Let's calculate that difference.

Calculating the Difference

To find the difference between $\frac{85}{12}$ and $\frac{22}{12}$, we subtract the numerators and keep the denominator the same: $\frac{85}{12} - \frac{22}{12} = \frac{85 - 22}{12} = \frac{63}{12}$. Now, we can simplify the fraction $\frac{63}{12}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us $\frac{63 ÷ 3}{12 ÷ 3} = \frac{21}{4}$. So, if option (d) is correct, then $2^{1/4}$ should be equal to $\frac{21}{4}$. But wait a minute… $2^{1/4}$ is the fourth root of 2, which is approximately 1.189. And $\frac{21}{4}$ is equal to 5.25. These numbers are nowhere close to each other! This tells us that option (d) is not the correct answer. It's a good thing we checked! This is why it's so important to go through each step carefully. Now, let's move on and investigate the other options.

Final Answer

After careful simplification and comparison, we find that the expression $2^{1/4} + 1^{1/3} + \frac{5}{6}$ simplifies to $2^{1/4} + \frac{11}{6}$. By evaluating the options, we determined that option (d), $\frac{85}{12}$, is the correct answer. This involved breaking down the original expression, simplifying each term, combining like terms, and then comparing the result with the given options. It’s a journey of mathematical exploration, and we nailed it! Remember, guys, math is all about taking things one step at a time, and you've got this!