Math Problem Statement
Solution
Let's analyze the expressions to determine if they are irrational numbers. A number is irrational if it cannot be written as a fraction of two integers, meaning it often involves non-repeating, non-terminating decimals like square roots of non-perfect squares.
Option 1:
- We can expand this expression using the distributive property (FOIL): Simplifying the square roots: = \sqrt{96} - \sqrt{24} + \sqrt{24} - \sqrt{6}$$ We know that $$\sqrt{96}, \sqrt{24}, \sqrt{6}$$ are all irrational numbers, meaning the result remains irrational.
Option 2:
- Again, expand this using FOIL: Simplifying the square roots: = \sqrt{144} + \sqrt{16} - \sqrt{36} - 2 = 12 + 4 - 6 - 2 = 8$$ Since the result is 8, this is a rational number.
Option 3:
- Rationalizing the denominator: = \frac{(\sqrt{6})^2 + 2\sqrt{6}\sqrt{3} + (\sqrt{3})^2}{(\sqrt{6})^2 - (\sqrt{3})^2}$$ Simplifying: $$= \frac{6 + 2\sqrt{18} + 3}{6 - 3} = \frac{9 + 6\sqrt{2}}{3} = 3 + 2\sqrt{2}$$ Since $$2\sqrt{2}$$ is irrational, the entire expression is irrational.
Option 4:
- Rationalizing the denominator: = \frac{(\sqrt{8})^2 + 2\sqrt{8}\sqrt{2} + (\sqrt{2})^2}{(\sqrt{8})^2 - (\sqrt{2})^2}$$ Simplifying: $$= \frac{8 + 2\sqrt{16} + 2}{8 - 2} = \frac{8 + 8 + 2}{6} = \frac{18}{6} = 3$$ Since the result is 3, this is a rational number.
Conclusion:
- Irrational numbers: Option 1 and Option 3.
Would you like further details on any of these steps?
Related questions:
- What is the significance of irrational numbers in algebra?
- How do you simplify expressions involving square roots?
- What is the difference between rationalizing the numerator vs. the denominator?
- How can you identify if a number is irrational without expanding expressions?
- How does the distributive property work when dealing with square roots?
Tip:
When simplifying square root expressions, always check if the roots can simplify to integers, which can help determine if the overall expression is rational or irrational.
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Math Problem Analysis
Mathematical Concepts
Irrational Numbers
Square Roots
Simplification
Rationalization
Formulas
(a + b)(c + d) = ac + ad + bc + bd
Rationalization of Denominators
Theorems
Properties of Square Roots
Distributive Property
Definition of Irrational Numbers
Suitable Grade Level
Grades 9-12
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