Math Problem Statement
Solution
Let's solve the problems from the uploaded image:
Problem 14: Simplify
-
We rewrite the cube root as: [ \sqrt[3]{\frac{25}{27}} = \frac{\sqrt[3]{25}}{\sqrt[3]{27}} ]
-
Evaluate each term:
- cannot be simplified further since 25 is not a perfect cube.
- (because ).
So, the simplified form is:
Problem 17: Simplify with no radicals in the denominator
-
To rationalize the denominator, multiply numerator and denominator by :
The simplified form is:
Would you like a step-by-step breakdown for any of these problems?
Here are 5 related questions to expand your understanding:
- How do you rationalize a denominator when it involves higher-order roots (e.g., cube roots)?
- What are some common strategies for simplifying radicals?
- How do you handle cube roots with variables in the numerator and denominator?
- Why do we rationalize the denominator in mathematical expressions?
- How can you simplify for practice?
Tip:
Always ensure the denominator is free of radicals by multiplying by the appropriate conjugate or root factor to simplify properly.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
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