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Chapter 7: Problem 3

A student incorrectly claimed that \(\sqrt{16}=8\). Evaluate \(\sqrt{16}\) correctly.

Short Answer

Expert verified
The correct evaluation is \( \sqrt{16} = 4 \).

Step by step solution

01

Understanding Square Roots

A square root of a number is a value that, when multiplied by itself, gives the number. For example, \(\text{if } x^2 = y\), then \( x = \sqrt{y}\).
02

Apply the Definition

We need to find a number that, when multiplied by itself, equals 16. Mathematically, this means finding \( x \text{ such that } x^2 = 16 \).
03

Find the Square Root of 16

\( \sqrt{16} \) means finding the number which squares to give 16. We check with possible numbers: \text{4} \text{ because } 4 \times 4 = 16. So, \sqrt{16} = 4 \.
04

Conclusion

Therefore, the correct evaluation of \( \sqrt{16} \) is 4, not 8. \( \sqrt{16} = 4 \. \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

square root definition
A square root is a number that produces a specified quantity when multiplied by itself. To better understand, let's consider the number 9. The square root of 9 is 3, because when 3 is multiplied by itself (3 脳 3), it equals 9. We can represent the square root of a number 'y' as \( \sqrt{y} \). In mathematical terms, if x squared equals y (x^2 = y), then x is the square root of y (x = \sqrt{y}). When visualizing square roots, imagine finding the side length of a square with a given area. This area should match the number under the square root.
evaluating square roots
To evaluate a square root, follow these steps:
  • Identify the number under the square root symbol.
  • Determine what number, when multiplied by itself, equals this number.
For instance, in the original exercise, we need to find the square root of 16. To solve this, we identify that 4 when multiplied by itself (4 脳 4), equals 16. Hence, \( \sqrt{16} = 4 \). Remember, evaluating square roots is about finding the correct value that squares back to the original number. Always double-check by squaring your answer to ensure it matches the original number.
properties of square roots
Square roots have interesting properties that can help you solve problems more easily:
  • The square root of a product: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
  • The square root of a quotient: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).
  • \texas Power Property: \( \sqrt{x^2} = |x| \) (absolute value of x).
  • \tNon-negative values: The principal square root is always non-negative. This means \( \sqrt{16} = 4 \) and not -4, even though both (4 脳 4) and (-4 脳 -4) result in 16.
Understanding these properties helps simplify multiplication or division involving square roots. They also play a key role in solving equations and inequalities involving square roots.
solving square root problems
When tackling square root problems, have these steps in mind:
  • Understand the problem statement clearly.
  • Apply your knowledge of square roots by using definitions and properties.
  • Break down the problem, like how we identified the incorrect claim in the exercise and then correctly evaluated \( \sqrt{16} \).
  • Ensure your solutions are logical and verify the results by squaring them to check if they match the original number.
If you face more complicated problems, consider breaking them into simpler parts or use algebraic techniques like isolating the square root on one side of the equation. Practicing these methods will sharpen your skills in solving square root problems accurately and efficiently.

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Most popular questions from this chapter

Simplify. Assume that all variables represent positive real numbers. \(-\sqrt[3]{27 t^{12}}\)

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