Math Problem Statement

Simplify the expressions in the image: 14. Simplify \(\sqrt[3]{\frac{25}{27}}\) and 17. Simplify \(\frac{3}{\sqrt{5}}\) without radicals in the denominator.

Solution

Let's solve the problems from the uploaded image:

Problem 14: Simplify 25273\sqrt[3]{\frac{25}{27}}

  1. We rewrite the cube root as: [ \sqrt[3]{\frac{25}{27}} = \frac{\sqrt[3]{25}}{\sqrt[3]{27}} ]

  2. Evaluate each term:

    • 253\sqrt[3]{25} cannot be simplified further since 25 is not a perfect cube.
    • 273=3\sqrt[3]{27} = 3 (because 33=273^3 = 27).

    So, the simplified form is: 2533\frac{\sqrt[3]{25}}{3}

Problem 17: Simplify 35\frac{3}{\sqrt{5}} with no radicals in the denominator

  1. To rationalize the denominator, multiply numerator and denominator by 5\sqrt{5}: 3555=355\frac{3}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5}

    The simplified form is: 355\frac{3\sqrt{5}}{5}

Would you like a step-by-step breakdown for any of these problems?


Here are 5 related questions to expand your understanding:

  1. How do you rationalize a denominator when it involves higher-order roots (e.g., cube roots)?
  2. What are some common strategies for simplifying radicals?
  3. How do you handle cube roots with variables in the numerator and denominator?
  4. Why do we rationalize the denominator in mathematical expressions?
  5. How can you simplify a2b53\sqrt[3]{a^2b^5} for practice?

Tip:

Always ensure the denominator is free of radicals by multiplying by the appropriate conjugate or root factor to simplify properly.

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Math Problem Analysis

Mathematical Concepts

Simplifying Radicals
Cube Roots
Rationalizing the Denominator

Formulas

Cube root simplification: \(\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}\)
Rationalizing the denominator: \(\frac{a}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} = \frac{a\sqrt{b}}{b}\)

Theorems

Properties of Radicals

Suitable Grade Level

Grades 8-10