Simplifying The Expression (750/512)^(1/3) A Step-by-Step Guide

This article delves into the simplification of the expression $\left(\frac{750}{512}\right)^{\frac{1}{3}}$. We will explore the fundamental concepts of exponents and radicals, prime factorization, and simplification techniques to arrive at the most reduced form of this expression. This comprehensive guide aims to provide a clear, step-by-step approach, ensuring readers understand each stage of the simplification process. Whether you're a student tackling algebra problems or simply seeking to enhance your mathematical skills, this article will offer valuable insights and practical methods for simplifying complex expressions.

Understanding the Basics: Exponents and Radicals

To simplify the expression $\left(\frac{750}{512}\right)^{\frac{1}{3}}$, we first need to understand the relationship between exponents and radicals. An exponent of $\frac{1}{3}$ signifies taking the cube root of the base. In mathematical terms, $x^{\frac{1}{3}} = \sqrt[3]{x}$. This fundamental concept is crucial for transforming the given expression into a more manageable form. The cube root of a number is a value that, when multiplied by itself three times, equals the original number. For example, the cube root of 8 is 2 because $2 \times 2 \times 2 = 8$. Grasping this concept is essential before we proceed with simplifying the fraction inside the parentheses. We need to break down both the numerator (750) and the denominator (512) into their prime factors to identify any perfect cubes. Perfect cubes are numbers that can be obtained by cubing an integer, such as 1, 8, 27, 64, 125, and so on. By recognizing perfect cubes within the prime factorization, we can effectively extract them from the cube root. Moreover, understanding the properties of exponents, such as $(a/b)^n = a^n / b^n$, allows us to distribute the exponent $\frac{1}{3}$ to both the numerator and the denominator separately. This makes the simplification process more organized and easier to follow. Therefore, a solid understanding of exponents and radicals forms the cornerstone of our approach to simplifying the given expression. Let's delve deeper into how we can apply these principles in the next sections.

Prime Factorization: Breaking Down 750 and 512

The next crucial step to simplify the expression involves prime factorization. We need to break down both 750 and 512 into their prime factors. Prime factorization is the process of expressing a number as a product of its prime factors, which are numbers that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11, etc.). Starting with 750, we can perform prime factorization as follows:

• 750 = 2 × 375
• 375 = 3 × 125
• 125 = 5 × 25
• 25 = 5 × 5

Thus, the prime factorization of 750 is $2 \times 3 \times 5^3$. Notice that $5^3$ is a perfect cube, which will be significant when we take the cube root. Now, let's factorize 512:

• 512 = 2 × 256
• 256 = 2 × 128
• 128 = 2 × 64
• 64 = 2 × 32
• 32 = 2 × 16
• 16 = 2 × 8
• 8 = 2 × 4
• 4 = 2 × 2

Therefore, the prime factorization of 512 is $2^9$, which can also be written as $(2^3)^3$. This shows that 512 is a perfect cube as well. The prime factorization process allows us to identify the building blocks of each number, revealing any perfect cube factors that can be extracted when taking the cube root. By expressing 750 and 512 in their prime factorized forms, we are setting the stage for the simplification of the original expression. In the next section, we will use these factorizations to simplify the expression $\left(\frac{750}{512}\right)^{\frac{1}{3}}$.

Applying the Cube Root: Simplifying the Expression

Now that we have the prime factorizations of 750 and 512, we can simplify the expression $\left(\frac{750}{512}\right)^{\frac{1}{3}}$. Recall that 750 = $2 \times 3 \times 5^3$ and 512 = $2^9$. Substituting these into the expression, we get:

$\left(\frac{2 \times 3 \times 5^3}{2^9}\right)^{\frac{1}{3}}$

We can simplify the fraction inside the parentheses by dividing both the numerator and the denominator by their common factors. In this case, both have factors of 2. Dividing the numerator by 2 and the denominator by $2$, we get:

$\left(\frac{3 \times 5^3}{2^8}\right)^{\frac{1}{3}}$

Now, we apply the exponent $\frac{1}{3}$ (which represents the cube root) to both the numerator and the denominator separately:

$\frac{(3 \times 5^3)^{\frac{1}{3}}}{(2^8)^{\frac{1}{3}}}$

Using the properties of exponents, we can distribute the exponent $\frac{1}{3}$:

$\frac{3^{\frac{1}{3}} \times (5^3)^{\frac{1}{3}}}{(2^8)^{\frac{1}{3}}}$

Now, we simplify each term. The cube root of $5^3$ is simply 5, as $(5^3)^{\frac{1}{3}} = 5$. For the denominator, we have $(2^8)^{\frac{1}{3}}$. We can rewrite $2^8$ as $2^{6} \times 2^2$, so the expression becomes:

$\frac{3^{\frac{1}{3}} \times 5}{(2^6 \times 2^2)^{\frac{1}{3}}}$

Distributing the exponent $\frac{1}{3}$ in the denominator:

$\frac{3^{\frac{1}{3}} \times 5}{(2^6)^{\frac{1}{3}} \times (2^2)^{\frac{1}{3}}}$

The cube root of $2^6$ is $2^2$ or 4, since $(2^6)^{\frac{1}{3}} = 2^{6 \times \frac{1}{3}} = 2^2 = 4$. Thus, the expression simplifies to:

$\frac{3^{\frac{1}{3}} \times 5}{4 \times (2^2)^{\frac{1}{3}}}$

We can rewrite $(2^2)^{\frac{1}{3}}$ as $4^{\frac{1}{3}}$, so we have:

$\frac{5 \times 3^{\frac{1}{3}}}{4 \times 4^{\frac{1}{3}}}$

Often, it is preferable to rationalize the denominator. However, in this case, leaving the expression in its current form is generally considered simplified, as further manipulation would not necessarily make it simpler. We have successfully simplified the original expression by applying the cube root and utilizing the properties of exponents and prime factorization. In the next section, we will summarize the steps and discuss the final result.

Final Result and Summary

After meticulously working through the steps of prime factorization and applying the cube root, we have successfully simplified the expression $\left(\frac{750}{512}\right)^{\frac{1}{3}}$. To recap, we started by understanding the relationship between exponents and radicals, recognizing that an exponent of $\frac{1}{3}$ is equivalent to taking the cube root. We then performed prime factorization on both 750 and 512, which yielded $750 = 2 \times 3 \times 5^3$ and $512 = 2^9$. Substituting these prime factorizations into the original expression, we obtained:

$\left(\frac{2 \times 3 \times 5^3}{2^9}\right)^{\frac{1}{3}}$

We simplified the fraction by dividing out the common factor of 2, resulting in:

$\left(\frac{3 \times 5^3}{2^8}\right)^{\frac{1}{3}}$

Applying the cube root to both the numerator and the denominator, we arrived at:

$\frac{(3 \times 5^3)^{\frac{1}{3}}}{(2^8)^{\frac{1}{3}}}$

Distributing the exponent and simplifying, we obtained:

$\frac{3^{\frac{1}{3}} \times 5}{(2^6 \times 2^2)^{\frac{1}{3}}} = \frac{5 \times 3^{\frac{1}{3}}}{4 \times 4^{\frac{1}{3}}}$

The final simplified form of the expression is:

$\frac{5 \times 3^{\frac{1}{3}}}{4 \times 4^{\frac{1}{3}}}$

This result is considered simplified because further manipulation would not necessarily lead to a more concise or easily understandable form. We have successfully extracted all perfect cube factors and applied the properties of exponents and radicals to achieve this simplified expression. Understanding these steps provides a solid foundation for simplifying other similar expressions in the future. This comprehensive approach demonstrates the power of prime factorization and exponent rules in simplifying complex mathematical expressions.