Which Statements Are True? Select The Five Correct Answers.A. 1.8 \textless 1.8 \sqrt{1.8}\ \textless \ 1.8 1.8 ​ \textless 1.8 B. 1.8 \textgreater 1 \sqrt{1.8}\ \textgreater \ 1 1.8 ​ \textgreater 1 C. 1.8 \textless 1.9 \sqrt{1.8}\ \textless \ \sqrt{1.9} 1.8 ​ \textless 1.9 ​ D. $1.3\ \textless \ \sqrt{1.8}\ \textless \

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In mathematics, square roots are an essential concept that deals with the number that, when multiplied by itself, gives a specified value. In this article, we will delve into the world of square roots and explore the truth behind five given statements. We will analyze each statement, using mathematical reasoning and calculations to determine which ones are true.

Statement A: 1.8 \textless 1.8\sqrt{1.8}\ \textless \ 1.8

To determine the truth behind statement A, we need to calculate the square root of 1.8. The square root of 1.8 is approximately 1.34164. Since 1.34164 is less than 1.8, statement A is TRUE.

Statement B: 1.8 \textgreater 1\sqrt{1.8}\ \textgreater \ 1

Next, we will examine statement B. To determine the truth behind this statement, we need to compare the square root of 1.8 with 1. As we calculated earlier, the square root of 1.8 is approximately 1.34164, which is greater than 1. Therefore, statement B is TRUE.

Statement C: 1.8 \textless 1.9\sqrt{1.8}\ \textless \ \sqrt{1.9}

Now, let's move on to statement C. To determine the truth behind this statement, we need to compare the square roots of 1.8 and 1.9. The square root of 1.8 is approximately 1.34164, and the square root of 1.9 is approximately 1.37228. Since 1.34164 is less than 1.37228, statement C is TRUE.

Statement D: 1.3 \textless 1.8 \textless 1.51.3\ \textless \ \sqrt{1.8}\ \textless \ 1.5

Next, we will examine statement D. To determine the truth behind this statement, we need to compare 1.3, the square root of 1.8, and 1.5. As we calculated earlier, the square root of 1.8 is approximately 1.34164. Since 1.3 is less than 1.34164, and 1.34164 is less than 1.5, statement D is TRUE.

Statement E: 1.8 \textless 1.7\sqrt{1.8}\ \textless \ \sqrt{1.7}

Finally, we will examine statement E. To determine the truth behind this statement, we need to compare the square roots of 1.8 and 1.7. The square root of 1.8 is approximately 1.34164, and the square root of 1.7 is approximately 1.30508. Since 1.30508 is less than 1.34164, statement E is FALSE.

Conclusion

In conclusion, we have analyzed five statements related to square roots and determined which ones are true. Statement A, B, C, and D are true, while statement E is false. By understanding the mathematical reasoning behind these statements, we can gain a deeper appreciation for the concept of square roots and their applications in mathematics.

Key Takeaways

  • The square root of a number is a value that, when multiplied by itself, gives the original number.
  • To determine the truth behind a statement, we need to calculate the square root of the given number and compare it with the other values.
  • Understanding the concept of square roots is essential in mathematics, and it has numerous applications in various fields.

Frequently Asked Questions

  • What is the square root of a number?
  • How do we calculate the square root of a number?
  • What are the applications of square roots in mathematics?

References

Glossary

  • Square Root: A value that, when multiplied by itself, gives the original number.
  • Mathematics: The study of numbers, quantities, and shapes.
  • Algebra: A branch of mathematics that deals with the study of variables and their relationships.
    Frequently Asked Questions About Square Roots =====================================================

In our previous article, we explored the truth behind five statements related to square roots. In this article, we will answer some of the most frequently asked questions about square roots.

Q: What is the square root of a number?

A: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

Q: How do we calculate the square root of a number?

A: There are several ways to calculate the square root of a number. One way is to use a calculator, which can quickly and accurately calculate the square root of a number. Another way is to use a mathematical formula, such as the Babylonian method, which involves making an initial guess and then repeatedly averaging the guess with the quotient of the number and the guess.

Q: What are the applications of square roots in mathematics?

A: Square roots have numerous applications in mathematics, including:

  • Algebra: Square roots are used to solve equations and inequalities, and to simplify expressions.
  • Geometry: Square roots are used to calculate the length of sides and diagonals of triangles and other geometric shapes.
  • Trigonometry: Square roots are used to calculate the values of trigonometric functions, such as sine and cosine.
  • Calculus: Square roots are used to calculate the derivatives and integrals of functions.

Q: What is the difference between a square root and a square?

A: A square root and a square are related but distinct concepts. A square is the result of multiplying a number by itself, while a square root is the value that, when multiplied by itself, gives the original number. For example, the square of 4 is 16, while the square root of 16 is 4.

Q: Can we have a negative square root?

A: In mathematics, we can have both positive and negative square roots. For example, the square root of 16 is both 4 and -4, because both 4 multiplied by 4 and -4 multiplied by -4 equal 16.

Q: How do we simplify square roots?

A: There are several ways to simplify square roots, including:

  • Factoring: We can factor the number under the square root sign into its prime factors, and then simplify the expression.
  • Rationalizing: We can rationalize the denominator of a fraction by multiplying the numerator and denominator by the conjugate of the denominator.
  • Using the Pythagorean theorem: We can use the Pythagorean theorem to simplify expressions involving square roots.

Q: What are some common mistakes to avoid when working with square roots?

A: Some common mistakes to avoid when working with square roots include:

  • Forgetting to simplify the expression: We should always simplify the expression under the square root sign before calculating the square root.
  • Not using the correct formula: We should use the correct formula for calculating the square root, such as the Babylonian method.
  • Not checking for negative square roots: We should always check for negative square roots, especially when working with equations and inequalities.

Conclusion

In conclusion, square roots are an essential concept in mathematics, with numerous applications in algebra, geometry, trigonometry, and calculus. By understanding the basics of square roots, we can simplify expressions, solve equations and inequalities, and calculate the length of sides and diagonals of triangles and other geometric shapes.

Key Takeaways

  • The square root of a number is a value that, when multiplied by itself, gives the original number.
  • We can calculate the square root of a number using a calculator or a mathematical formula.
  • Square roots have numerous applications in mathematics, including algebra, geometry, trigonometry, and calculus.
  • We can simplify square roots by factoring, rationalizing, and using the Pythagorean theorem.
  • We should avoid common mistakes, such as forgetting to simplify the expression, not using the correct formula, and not checking for negative square roots.

References

Glossary

  • Square Root: A value that, when multiplied by itself, gives the original number.
  • Mathematics: The study of numbers, quantities, and shapes.
  • Algebra: A branch of mathematics that deals with the study of variables and their relationships.
  • Geometry: A branch of mathematics that deals with the study of shapes and their properties.
  • Trigonometry: A branch of mathematics that deals with the study of triangles and their properties.