What Is The Simplified Form Of The Following Expression? Assume $y \neq 0$.$\sqrt[3]{\frac{12 X^2}{16 Y}}$

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the simplified form of the expression $\sqrt[3]{\frac{12 x^2}{16 y}}$, assuming that $y \neq 0$. We will break down the expression into manageable parts, apply the necessary rules and formulas, and arrive at the simplified form.

Understanding the Expression

The given expression is a radical expression, which involves the cube root of a fraction. The cube root is denoted by the symbol $\sqrt[3]{\ }$, and it represents the number that, when multiplied by itself three times, gives the original number. In this case, we have $\sqrt[3]{\frac{12 x^2}{16 y}}$, which means we need to find the cube root of the fraction $\frac{12 x^2}{16 y}$.

Breaking Down the Fraction

To simplify the expression, we need to break down the fraction into its prime factors. The numerator is $12 x^2$, and the denominator is $16 y$. We can factorize both numbers as follows:

  • 12=22β‹…312 = 2^2 \cdot 3

  • 16=2416 = 2^4

  • x2=x2x^2 = x^2

  • y=yy = y

Now, we can rewrite the fraction as:

12x216y=22β‹…3β‹…x224β‹…y\frac{12 x^2}{16 y} = \frac{2^2 \cdot 3 \cdot x^2}{2^4 \cdot y}

Applying the Rules of Exponents

To simplify the expression further, we need to apply the rules of exponents. Specifically, we need to use the rule that states:

aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}

where $a$ is a positive number, and $m$ and $n$ are integers. In this case, we have:

2224=22βˆ’4=2βˆ’2\frac{2^2}{2^4} = 2^{2-4} = 2^{-2}

Similarly, we can simplify the other parts of the fraction:

31=3\frac{3}{1} = 3

x21=x2\frac{x^2}{1} = x^2

1y=1y\frac{1}{y} = \frac{1}{y}

Now, we can rewrite the fraction as:

12x216y=22β‹…3β‹…x224β‹…y=2βˆ’2β‹…3β‹…x2β‹…1y\frac{12 x^2}{16 y} = \frac{2^2 \cdot 3 \cdot x^2}{2^4 \cdot y} = 2^{-2} \cdot 3 \cdot x^2 \cdot \frac{1}{y}

Simplifying the Cube Root

Now that we have simplified the fraction, we can take the cube root of the expression. To do this, we need to apply the rule that states:

am3=am3\sqrt[3]{a^m} = a^{\frac{m}{3}}

where $a$ is a positive number, and $m$ is an integer. In this case, we have:

2βˆ’2β‹…3β‹…x2β‹…1y3=2βˆ’23β‹…313β‹…x23β‹…1y13\sqrt[3]{2^{-2} \cdot 3 \cdot x^2 \cdot \frac{1}{y}} = 2^{-\frac{2}{3}} \cdot 3^{\frac{1}{3}} \cdot x^{\frac{2}{3}} \cdot \frac{1}{y^{\frac{1}{3}}}

Final Simplified Form

After applying the rules of exponents and simplifying the cube root, we arrive at the final simplified form of the expression:

12x216y3=2βˆ’23β‹…313β‹…x23β‹…1y13\sqrt[3]{\frac{12 x^2}{16 y}} = 2^{-\frac{2}{3}} \cdot 3^{\frac{1}{3}} \cdot x^{\frac{2}{3}} \cdot \frac{1}{y^{\frac{1}{3}}}

Conclusion

In this article, we have simplified the expression $\sqrt[3]{\frac{12 x^2}{16 y}}$, assuming that $y \neq 0$. We broke down the fraction into its prime factors, applied the rules of exponents, and simplified the cube root to arrive at the final simplified form. This process demonstrates the importance of understanding the rules of exponents and simplifying radical expressions in mathematics.

Additional Tips and Resources

  • To simplify radical expressions, it is essential to understand the rules of exponents and prime factorization.
  • When simplifying cube roots, remember to apply the rule $\sqrt[3]{a^m} = a^{\frac{m}{3}}$.
  • Practice simplifying radical expressions with different numbers and variables to become proficient in this skill.
  • For more information on simplifying radical expressions, consult a mathematics textbook or online resource.

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Mathematics for the Nonmathematician" by Morris Kline
  • [3] "Simplifying Radical Expressions" by Math Open Reference

Glossary

  • Cube root: The number that, when multiplied by itself three times, gives the original number.
  • Exponent: A number that represents the power to which a base number is raised.
  • Prime factorization: The process of breaking down a number into its prime factors.
  • Radical expression: An expression that involves a root, such as a square root or cube root.
    Frequently Asked Questions: Simplifying Radical Expressions ===========================================================

Q: What is the simplified form of the expression $\sqrt[3]{\frac{12 x^2}{16 y}}$?

A: The simplified form of the expression $\sqrt[3]{\frac{12 x^2}{16 y}}$ is $2^{-\frac{2}{3}} \cdot 3^{\frac{1}{3}} \cdot x^{\frac{2}{3}} \cdot \frac{1}{y^{\frac{1}{3}}}$.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to break down the fraction into its prime factors, apply the rules of exponents, and simplify the cube root.

Q: What is the rule for simplifying cube roots?

A: The rule for simplifying cube roots is $\sqrt[3]{a^m} = a^{\frac{m}{3}}$, where $a$ is a positive number, and $m$ is an integer.

Q: How do I apply the rule of exponents to simplify a radical expression?

A: To apply the rule of exponents, you need to use the rule $\frac{am}{an} = a^{m-n}$, where $a$ is a positive number, and $m$ and $n$ are integers.

Q: What is the importance of understanding the rules of exponents and simplifying radical expressions?

A: Understanding the rules of exponents and simplifying radical expressions is essential in mathematics, as it helps you to simplify complex expressions and solve problems more efficiently.

Q: How can I practice simplifying radical expressions?

A: You can practice simplifying radical expressions by using online resources, such as math websites and apps, or by working on problems in a mathematics textbook.

Q: What are some common mistakes to avoid when simplifying radical expressions?

A: Some common mistakes to avoid when simplifying radical expressions include:

  • Not breaking down the fraction into its prime factors
  • Not applying the rules of exponents correctly
  • Not simplifying the cube root correctly

Q: How can I check my work when simplifying radical expressions?

A: You can check your work by plugging the simplified expression back into the original equation and verifying that it is true.

Q: What are some real-world applications of simplifying radical expressions?

A: Simplifying radical expressions has many real-world applications, including:

  • Calculating distances and heights in geometry and trigonometry
  • Solving problems in physics and engineering
  • Working with financial and economic data

Q: How can I use technology to simplify radical expressions?

A: You can use technology, such as calculators and computer software, to simplify radical expressions and check your work.

Q: What are some additional resources for learning about simplifying radical expressions?

A: Some additional resources for learning about simplifying radical expressions include:

  • Mathematics textbooks and online resources
  • Online tutorials and video lessons
  • Math apps and software

Conclusion

Simplifying radical expressions is an essential skill in mathematics, and it has many real-world applications. By understanding the rules of exponents and simplifying radical expressions, you can solve problems more efficiently and effectively. Remember to practice regularly and use technology to check your work. With practice and patience, you can become proficient in simplifying radical expressions and tackle complex problems with confidence.

Glossary

  • Cube root: The number that, when multiplied by itself three times, gives the original number.
  • Exponent: A number that represents the power to which a base number is raised.
  • Prime factorization: The process of breaking down a number into its prime factors.
  • Radical expression: An expression that involves a root, such as a square root or cube root.
  • Rule of exponents: A set of rules that govern the behavior of exponents in mathematical expressions.
  • Simplifying radical expressions: The process of simplifying complex expressions that involve roots.