Simplify Cube Roots: Math Problem Solved!
Hey math enthusiasts! Let's dive into a cool cube root problem that's super fun to solve. We're going to break down the expression $(7 \cdot 250)^{\frac{1}{3}}$ and find out which of the given options is equal to it. This kind of problem is a great way to sharpen those math skills and understand how exponents and roots work together. So, grab your pencils, and let's get started! We will explore a step-by-step method to simplify the cube root of the given expression, focusing on prime factorization and the properties of exponents.
Understanding the Problem: The Cube Root Challenge
Alright, so we're looking at $(7 \cdot 250)^{\frac{1}{3}}$. What does this even mean? Well, the expression inside the parentheses is $7 \cdot 250$, and the power of $\frac{1}{3}$ tells us we need to find the cube root of the result. Think of it like this: we need to find a number that, when multiplied by itself three times, equals $7 \cdot 250$. This might seem tricky at first, but with a bit of prime factorization and the properties of exponents, we can crack it! The key here is to simplify the number under the cube root, which helps us identify any perfect cubes that can be extracted. The goal is to rewrite the expression in a way that makes it easier to find the cube root.
Before we start, let's quickly review what a cube root is. The cube root of a number $x$, denoted as $\sqrt[3]{x}$, is a value that, when multiplied by itself three times, gives $x$. For example, the cube root of 8 is 2 because $2 \cdot 2 \cdot 2 = 8$. Now, let's get back to our problem. To simplify $(7 \cdot 250)^{\frac{1}{3}}$, we first need to break down the number inside the cube root into its prime factors. This will help us identify any perfect cubes and simplify the expression. Breaking down the number into prime factors is a fundamental step in simplifying radicals, and it often involves dividing the number by the smallest prime numbers until we're left with only prime factors. Let's see how this works in our case. It's like a puzzle where we have to find the building blocks (prime factors) that make up the whole number. Once we have the prime factors, we can regroup them to identify any perfect cubes and simplify the expression further. The prime factorization will make the process of finding the cube root much easier. Trust me, it's not as scary as it sounds! It's all about breaking things down into their simplest parts.
Step-by-Step Solution: Unraveling the Cube Root
Here’s how we'll break down the problem step by step to find the correct answer:
Step 1: Prime Factorization
First, let's break down the numbers in the expression. We have 7, which is already a prime number, and 250. Let's factorize 250:
So, the prime factorization of 250 is $2 \cdot 5 \cdot 5 \cdot 5$. Now, let's put it all together. Our original expression $(7 \cdot 250)^{\frac{1}{3}}$ becomes $(7 \cdot 2 \cdot 5 \cdot 5 \cdot 5)^{\frac{1}{3}}$ or $(7 \cdot 2 \cdot 53){\frac{1}{3}}$.
Step 2: Simplifying the Expression
Now we have $(7 \cdot 2 \cdot 53){\frac{1}{3}}$. Remember, the cube root of a product is the product of the cube roots. We can rewrite the expression as $\sqrt[3]{7 \cdot 2 \cdot 5^3}$. We know that $\sqrt[3]{5^3} = 5$, so we can take the 5 out of the cube root. This gives us $5 \cdot \sqrt[3]{7 \cdot 2}$. Simplifying further, we get $5 \cdot \sqrt[3]{14}$.
Step 3: Matching with the Options
Let’s look back at our options:
A. $15 \cdot \sqrt[3]{2}$ B. $\sqrt[3]{30}$ C. $\sqrt[3]{10}$ D. $5 \cdot \sqrt[3]{2}$
Our simplified expression is $5 \cdot \sqrt[3]{14}$. None of the options exactly match this, which means there might be a trick! Reviewing our steps, we see that we factored 250 into $2 \cdot 5^3$ and multiplied it by 7. Therefore, our expression is equal to $(7 \cdot 2 \cdot 53){\frac{1}{3}}$, which simplifies to $5 \cdot \sqrt[3]{14}$. There might be a typo in the options. However, let's analyze each option provided.
Analyzing the Answers: Finding the Correct Match
Now, let's analyze the multiple-choice options we were given to see which one best fits our simplified expression. This is where we determine if any of the given answers are equivalent to the simplified form of our original expression. This step requires careful consideration of the properties of cube roots and exponents.
Option A: $15 \cdot \sqrt[3]{2}$
This option simplifies to $3 \cdot 5 \cdot \sqrt[3]{2}$. To get this from our original expression, we would need to have factors of 3 and 2 inside the cube root. However, based on our calculations, we do not have a 3 in our prime factorization. So, this option is incorrect.
Option B: $\sqrt[3]{30}$
This option simplifies to $\sqrt[3]{2 \cdot 3 \cdot 5}$. Comparing this with our prime factors ($7 \cdot 2 \cdot 5^3$), we can see that these expressions are not the same. Therefore, option B is incorrect as well.
Option C: $\sqrt[3]{10}$
This option equals $\sqrt[3]{2 \cdot 5}$. This also does not align with our prime factors. Thus, this option is also not the correct answer.
Option D: $5 \cdot \sqrt[3]{2}$
Our simplified expression is $5 \cdot \sqrt[3]{14}$. Comparing this to option D, we see that it is not equal, as it doesn’t have the factor 7 inside the cube root. So, Option D is also incorrect, because our simplified answer has a cube root of 14, not 2. However, if there was a typo in the provided options and one of the options had $5 \cdot \sqrt[3]{14}$, then that would be the correct choice. As we can't find an exact match, the question might have a typo, but based on our analysis, we can arrive at the closest possible answer. Based on the prime factorization of the original expression, the most appropriate answer is the one that contains factors 2, 5 and 7 under the cube root, even if there's a minor error in the options.
Conclusion: The Final Answer
After breaking down the expression and carefully analyzing the options, we can see that none of the answers exactly match our simplified form of $5 \cdot \sqrt[3]{14}$. Based on the prime factorization, the closest possible match would be an answer that contains the factors of 2, 5 and 7 under the cube root, even if there's a minor error in the options. This process shows us how to use prime factorization and exponent properties to simplify and solve cube root problems. Keep practicing, and you'll become a cube root master in no time! Remember, the key is to break down the problem into smaller, manageable steps. And always double-check your work!