Ordering Numbers With Roots: A Math Problem Solved

Hey guys! Let's dive into a cool math problem today that involves ordering numbers with roots. It might seem a bit tricky at first, but we'll break it down step by step so it's super easy to understand. We're given three numbers: $x = \sqrt[3]{5^2}$, $y = \sqrt{3^2}$, and $z = \sqrt[4]{25}$. Our mission is to figure out the correct order of these numbers from the smallest to the largest. So, grab your thinking caps, and let’s get started!

Understanding the Problem

Before we jump into solving, it's crucial to really understand what the problem is asking. We have three numbers, each expressed with different types of roots – a cube root, a square root, and a fourth root. The key here is that we can't directly compare them in their current form. To make a fair comparison, we need to express them in a way that we can easily relate to each other. This usually means either getting rid of the roots or making the roots the same. Think of it like comparing apples and oranges; we need a common unit to measure them against. This initial step of understanding sets the stage for a smooth solution. Understanding the problem thoroughly will prevent us from making hasty decisions and guide us towards the most efficient solution path. Remember, in math, a clear understanding is half the battle won! We will explore how to manipulate these roots and exponents to bring them to a comparable form.

Converting Roots to Exponents

The first smart move in tackling this problem is to convert the roots into exponents. Why? Because exponents are often easier to manipulate and compare. Remember this cool trick: the ${n}$-th root of a number is the same as raising that number to the power of ${\frac{1}{n}}$. So, we can rewrite our numbers like this:

• ${x = \sqrt[3]{5^2} = 5^{\frac{2}{3}}}$
• ${y = \sqrt{3^2} = 3^{\frac{2}{2}} = 3^1 = 3}$
• ${z = \sqrt[4]{25}}$. Now, here's a little extra step for ${z}$. Notice that 25 is ${5^2}$, so we can rewrite ${z}$ as ${z = \sqrt[4]{5^2} = 5^{\frac{2}{4}} = 5^{\frac{1}{2}}}$

Now we have ${x = 5^{\frac{2}{3}}}$, ${y = 3}$, and ${z = 5^{\frac{1}{2}}}$. See? Things are already looking clearer! Converting roots to exponents is a fundamental technique in algebra and calculus. It allows us to apply exponent rules, which are often simpler to work with than root rules. By expressing our numbers in exponential form, we've essentially translated the problem into a different language that's easier for us to handle. This step is a game-changer because it lays the groundwork for the next phase, where we'll make the exponents comparable. This conversion is not just a mathematical trick; it's a way of seeing the problem from a new angle, which is a valuable skill in problem-solving.

Finding a Common Exponent

Okay, we've got our numbers in exponential form: ${x = 5^{\frac{2}{3}}}$, ${y = 3}$, and ${z = 5^{\frac{1}{2}}}$. To compare ${x}$ and ${z}$, it would be super helpful if they had the same exponent. To do this, we need to find a common denominator for the fractions ${\frac{2}{3}}$ and ${\frac{1}{2}}$. The least common multiple of 3 and 2 is 6, so let's rewrite the exponents with a denominator of 6:

• ${x = 5^{\frac{2}{3}} = 5^{\frac{4}{6}}}$
• ${z = 5^{\frac{1}{2}} = 5^{\frac{3}{6}}}$

Now we can see that ${x = 5^{\frac{4}{6}}}$ and ${z = 5^{\frac{3}{6}}}$ are much easier to compare. Finding a common exponent is a clever strategy because it allows us to compare the bases directly. If two exponential expressions have the same exponent, then the expression with the larger base will be larger overall. In our case, we've made the exponents of ${x}$ and ${z}$ the same, which means we can now focus on the bases. This step demonstrates a key principle in mathematics: transforming a problem into a more manageable form. By rewriting the fractions with a common denominator, we've created a level playing field for comparison. This technique is not only useful for comparing numbers but also for solving equations and simplifying expressions.

Comparing x and z

With ${x = 5^{\frac{4}{6}}}$ and ${z = 5^{\frac{3}{6}}}$, we're in a great spot to compare them. Since the base (5) is greater than 1, a larger exponent means a larger number. So, because ${\frac{4}{6} > \frac{3}{6}}$, we know that ${5^{\frac{4}{6}} > 5^{\frac{3}{6}}}$ which means ${x > z}$. Awesome! We've got our first piece of the puzzle. Comparing x and z after finding a common exponent is straightforward. The principle here is that for a base greater than 1, the exponential function is increasing. This means that as the exponent increases, the value of the expression also increases. It's a fundamental concept in understanding exponential growth. This comparison step is a perfect example of how strategic manipulation of numbers can lead to a clear and simple conclusion. By making the exponents the same, we were able to isolate the effect of the base on the value of the expression, making the comparison direct and intuitive. This approach highlights the power of mathematical tools in simplifying complex problems.

Comparing with y

Now, we need to bring ${y = 3}$ into the mix. This is where things get a little interesting. We need to compare ${y}$ with ${x = 5^{\frac{4}{6}}}$ and ${z = 5^{\frac{3}{6}}}$. Let’s think about ${z = 5^{\frac{3}{6}} = 5^{\frac{1}{2}} = \sqrt{5}}$. We know that ${\sqrt{4} = 2}$ and ${\sqrt{9} = 3}$, so ${\sqrt{5}}$ is somewhere between 2 and 3. This tells us that ${z < 3}$, so ${z < y}$.

Next, let’s tackle ${x = 5^{\frac{4}{6}}}$. We can rewrite this as ${x = (5^4)^{\frac{1}{6}} = 625^{\frac{1}{6}}}$. Now, we need to estimate the sixth root of 625. Since ${2^6 = 64}$ and ${3^6 = 729}$, we know that (

\sqrt[6]{625}) is between 2 and 3. However, 625 is much closer to 729 than 64, so (

\sqrt[6]{625}) is closer to 3 than 2. But is it greater or smaller than 3? This is a tricky part. Since 3^6 is bigger than 625, the sixth root of 625 will be smaller than 3. Thus, x < 3.

Comparing with y, which equals 3, requires a bit more finesse. We've established benchmarks and now need to refine our estimates. The realization that is less than 3 is crucial, placing x in context relative to y. This step exemplifies how number sense and estimation are valuable tools in mathematical reasoning. It's not always about finding the exact answer, but understanding the relative magnitudes. This part of the problem challenges us to think critically about the relationships between numbers and roots, pushing us beyond rote calculations. This comparison highlights the importance of approximation and estimation in mathematical problem-solving, especially when dealing with irrational numbers.

Putting It All Together

Okay, we've done the legwork! Let's recap what we've found:

• We know ${x > z}$
• We figured out ${z < y}$
• We determined ${x < y}$

Putting it all together, the correct order from smallest to largest is ${z < x < y}$. Putting it all together is the final step in our mathematical journey. It's where we consolidate all the individual pieces of information we've gathered and assemble them into a coherent solution. This step requires a clear understanding of the relationships we've established along the way. In our case, we've systematically compared each number to the others, and now we can confidently state the correct order. This synthesis is not just about stating the answer; it's about demonstrating a complete understanding of the problem-solving process. It reinforces the logical flow of our reasoning and highlights the interconnectedness of the different steps we've taken. This final act of synthesis is a testament to the power of structured thinking in mathematics.

Final Answer

So, the final answer is that the numbers in order from smallest to largest are ${z < x < y}$. You did it! This problem was a great workout for our brains, taking us through converting roots to exponents, finding common exponents, and comparing numbers. Keep practicing these skills, and you'll become a math whiz in no time! The final answer is the culmination of our efforts, but the real value lies in the journey we took to get there. We've not only solved a problem but also honed our mathematical skills and deepened our understanding of key concepts. This problem, with its mix of roots, exponents, and comparisons, is a microcosm of the challenges and rewards of mathematical thinking. By breaking it down into manageable steps, we've demonstrated that even complex problems can be tackled with the right approach. This final solution is not just an answer; it's a testament to our problem-solving abilities and a stepping stone to future mathematical adventures. Remember, every problem solved is a victory earned, and every challenge overcome makes us stronger mathematicians.