Math Problem Statement
Which of the following contains all of the irrational numbers in the set?
A.) negative root index blank of 7 comma end root short dash 0.7 with bar on top comma space square root of 25
B.) short dash 0.7 with bar on top comma space 1.5 comma space square root of 25
C.) short dash square root of 7 comma space square root of 25
D.) short dash square root of 7
Solution
To determine which set contains all the irrational numbers, let's analyze each element:
Step 1: Understand Irrational Numbers
An irrational number is a number that cannot be expressed as a fraction , where and are integers and . It has a non-terminating, non-repeating decimal expansion. Examples include numbers like or .
Step 2: Analyze the Options
Option A:
- : Irrational (square root of a non-perfect square).
- : Rational (repeating decimal, equal to ).
- : Rational (perfect square root).
Contains 1 irrational number: .
Option B:
- : Rational.
- : Rational (finite decimal, equal to ).
- : Rational.
Contains no irrational numbers.
Option C:
- : Irrational.
- : Rational.
Contains 1 irrational number: .
Option D:
- : Irrational.
Contains 1 irrational number: .
Step 3: Correct Answer
The question asks for the set containing all the irrational numbers in the list. Since is the only irrational number present across all sets, the correct answer is D.
Do you want details or have any questions?
Here are 5 questions to expand understanding:
- What distinguishes rational and irrational numbers in their decimal representation?
- Can irrational numbers be expressed as approximations?
- Why is considered rational?
- Are all roots of non-perfect squares irrational?
- How do repeating decimals classify as rational numbers?
Tip: Memorize that any square root of a perfect square (e.g., ) is always rational.
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Math Problem Analysis
Mathematical Concepts
Rational and Irrational Numbers
Properties of Square Roots
Formulas
Square root operation
Definition of rational and irrational numbers
Theorems
A square root of a non-perfect square is irrational
Decimal representation of rational numbers (terminating or repeating)
Suitable Grade Level
Grades 8-10