What Is The Simplified Form Of $\sqrt{\frac{72 X^{16}}{50 X^{36}}}$? Assume $x \neq 0$.A. $\frac{6}{5 X^{10}}$B. $ 6 5 X 2 \frac{6}{5 X^2} 5 X 2 6 ​ [/tex]C. $\frac{6}{5} X^{10}$D. $\frac{6}{5} X^2$

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Introduction

Radical expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will delve into the world of radical expressions and explore the simplified form of the given expression: $\sqrt{\frac{72 x^{16}}{50 x^{36}}}$.

Understanding the Problem

The given expression involves a square root, which can be simplified by using the properties of radicals. To simplify the expression, we need to apply the rules of exponents and radicals. The expression can be rewritten as:

72x1650x36=72x1650x36\sqrt{\frac{72 x^{16}}{50 x^{36}}} = \frac{\sqrt{72 x^{16}}}{\sqrt{50 x^{36}}}

Simplifying the Numerator and Denominator

To simplify the expression, we need to simplify the numerator and denominator separately. Let's start with the numerator:

72x16=23⋅32⋅x16\sqrt{72 x^{16}} = \sqrt{2^3 \cdot 3^2 \cdot x^{16}}

Using the property of radicals, we can rewrite the expression as:

23⋅32⋅x16=2⋅3⋅x8=6x8\sqrt{2^3 \cdot 3^2 \cdot x^{16}} = 2 \cdot 3 \cdot x^8 = 6 x^8

Now, let's simplify the denominator:

50x36=2⋅52⋅x36\sqrt{50 x^{36}} = \sqrt{2 \cdot 5^2 \cdot x^{36}}

Using the property of radicals, we can rewrite the expression as:

2⋅52⋅x36=5⋅x18=5x18\sqrt{2 \cdot 5^2 \cdot x^{36}} = 5 \cdot x^{18} = 5 x^{18}

Combining the Simplified Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to get the final simplified expression:

72x1650x36=6x85x18\frac{\sqrt{72 x^{16}}}{\sqrt{50 x^{36}}} = \frac{6 x^8}{5 x^{18}}

Applying the Quotient Rule of Exponents

To simplify the expression further, we can apply the quotient rule of exponents, which states that:

xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}

Using this rule, we can rewrite the expression as:

6x85x18=65x8−18=65x−10\frac{6 x^8}{5 x^{18}} = \frac{6}{5} x^{8-18} = \frac{6}{5} x^{-10}

Simplifying the Negative Exponent

To simplify the negative exponent, we can rewrite the expression as:

65x−10=65⋅1x10=65x10\frac{6}{5} x^{-10} = \frac{6}{5} \cdot \frac{1}{x^{10}} = \frac{6}{5 x^{10}}

Conclusion

In conclusion, the simplified form of the given expression is:

65x10\frac{6}{5 x^{10}}

This expression can be verified by plugging in values for xx and checking if the expression holds true.

Answer

The correct answer is:

A. $\frac{6}{5 x^{10}}$

Discussion

This problem requires a good understanding of radical expressions and the properties of exponents. The key to solving this problem is to simplify the numerator and denominator separately and then combine them to get the final simplified expression. The quotient rule of exponents is also an essential tool in simplifying the expression.

Tips and Tricks

  • When simplifying radical expressions, it's essential to apply the properties of radicals and exponents.
  • The quotient rule of exponents is a powerful tool in simplifying expressions involving fractions.
  • Negative exponents can be simplified by rewriting the expression as a fraction.

Practice Problems

  • Simplify the expression: $\sqrt{\frac{144 x^{24}}{36 x^{40}}}$
  • Simplify the expression: $\sqrt{\frac{9 x^{32}}{81 x^{48}}}$

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
    Simplifying Radical Expressions: A Q&A Guide =====================================================

Introduction

Radical expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will delve into the world of radical expressions and provide a Q&A guide to help you master the art of simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is an expression that contains a square root or a higher-order root. It is denoted by the symbol \sqrt{}.

Q: What are the properties of radicals?

A: The properties of radicals include:

  • The product rule: ab=a⋅b\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}
  • The quotient rule: ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}
  • The power rule: am=am2\sqrt{a^m} = a^{\frac{m}{2}}

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to apply the properties of radicals and the rules of exponents. Here are the steps:

  1. Simplify the numerator and denominator separately.
  2. Apply the product rule, quotient rule, and power rule as needed.
  3. Combine the simplified numerator and denominator to get the final simplified expression.

Q: What is the quotient rule of exponents?

A: The quotient rule of exponents states that:

xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}

This rule can be used to simplify expressions involving fractions.

Q: How do I handle negative exponents?

A: To handle negative exponents, you can rewrite the expression as a fraction. For example:

1xm=x−m\frac{1}{x^m} = x^{-m}

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

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

  • Not applying the properties of radicals and exponents correctly.
  • Not simplifying the numerator and denominator separately.
  • Not combining the simplified numerator and denominator correctly.

Q: How can I practice simplifying radical expressions?

A: You can practice simplifying radical expressions by working through examples and exercises. Here are some tips:

  • Start with simple expressions and gradually move on to more complex ones.
  • Use online resources and practice problems to help you master the art of simplifying radical expressions.
  • Review the properties of radicals and exponents regularly to ensure that you remember them.

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 equations and inequalities in algebra and calculus.
  • Working with complex numbers and polynomials in number theory and algebraic geometry.

Conclusion

Simplifying radical expressions is a crucial skill for any math enthusiast. By mastering the art of simplifying radical expressions, you can solve a wide range of problems in algebra, geometry, and other areas of mathematics. Remember to apply the properties of radicals and exponents correctly, and to practice regularly to ensure that you remember them.

Practice Problems

  • Simplify the expression: $\sqrt{\frac{144 x^{24}}{36 x^{40}}}$
  • Simplify the expression: $\sqrt{\frac{9 x^{32}}{81 x^{48}}}$
  • Simplify the expression: $\sqrt{\frac{16 x^{36}}{64 x^{60}}}$

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton