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 pq\frac{p}{q}, where pp and qq are integers and q0q \neq 0. It has a non-terminating, non-repeating decimal expansion. Examples include numbers like 7\sqrt{7} or π\pi.


Step 2: Analyze the Options

Option A:

7,  0.77,  25-\sqrt{7},\; -0.7\overline{7},\; \sqrt{25}

  • 7-\sqrt{7}: Irrational (square root of a non-perfect square).
  • 0.77-0.7\overline{7}: Rational (repeating decimal, equal to 79-\frac{7}{9}).
  • 25=5\sqrt{25} = 5: Rational (perfect square root).

Contains 1 irrational number: 7-\sqrt{7}.


Option B:

0.77,  1.5,  25-0.7\overline{7},\; 1.5,\; \sqrt{25}

  • 0.77-0.7\overline{7}: Rational.
  • 1.51.5: Rational (finite decimal, equal to 32\frac{3}{2}).
  • 25=5\sqrt{25} = 5: Rational.

Contains no irrational numbers.


Option C:

7,  25-\sqrt{7},\; \sqrt{25}

  • 7-\sqrt{7}: Irrational.
  • 25=5\sqrt{25} = 5: Rational.

Contains 1 irrational number: 7-\sqrt{7}.


Option D:

7-\sqrt{7}

  • 7-\sqrt{7}: Irrational.

Contains 1 irrational number: 7-\sqrt{7}.


Step 3: Correct Answer

The question asks for the set containing all the irrational numbers in the list. Since 7-\sqrt{7} 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:

  1. What distinguishes rational and irrational numbers in their decimal representation?
  2. Can irrational numbers be expressed as approximations?
  3. Why is 25\sqrt{25} considered rational?
  4. Are all roots of non-perfect squares irrational?
  5. How do repeating decimals classify as rational numbers?

Tip: Memorize that any square root of a perfect square (e.g., 9,16\sqrt{9}, \sqrt{16}) is always rational.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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