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Chapter 5: Problem 3

Evaluate each of the following. For example, \(\sqrt{25}=5\). \(-\sqrt{100}\)

Short Answer

Expert verified
The result is -10.

Step by step solution

01

Identify the Radicand

The given expression is \( -\sqrt{100} \). Here, the radicand, which is the number inside the square root, is 100.
02

Calculate the Principal Square Root

Find the principal (positive) square root of 100. Since 10 multiplied by itself equals 100, \( \sqrt{100} = 10 \).
03

Apply the Negative Sign

The expression includes a negative sign in front of the square root. This means we need to multiply the square root by -1. Therefore, the expression \(-\sqrt{100}\) evaluates to \(-10\).

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Key Concepts

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

Understanding the Radicand
In mathematics, when dealing with square roots, the radicand is a crucial element to grasp. The radicand is simply the number that is placed under the square root symbol. For example, in the expression \(\sqrt{100}\), the radicand is 100. It is the value for which we seek the root and thus forms the foundation of the square root process.
  • The radicand must remain non-negative when dealing with real square roots, since the square root of a negative number isn't defined in the real number system.
  • Understanding the radicand helps in accurately determining which number multiplied by itself gives us the radicand. In our example with \(\sqrt{100}\), we are looking for a number which, when squared, equals 100.
The concept of a radicand is straightforward but essential for correctly evaluating square root problems. Knowing what the radicand is allows us to move on to finding its square root.
Discovering the Principal Square Root
The principal square root is the positive square root of a number. Whenever you see a square root symbol, it typically implies the principal square root, unless otherwise specified. In the square root expression \(\sqrt{100}\), the principal square root is 10, because 10 is the positive number that, when multiplied by itself, equals 100.
  • Remember, the principal square root will always be non-negative. It's important to distinguish this, especially when negative signs appear in expressions.
  • Principal square roots are foundational for solving equations and understanding how to manipulate expressions involving square roots.
By determining the principal square root, you identify the primary result of the square root operation. Once the principal square root is found, additional operations, such as applying a negative sign, can be performed.
Negative Sign Application in Square Roots
In mathematical expressions, applying a negative sign to the square root is a simple yet important step. The presence of a negative sign in front of the square root indicates that we should take the principal square root and then multiply it by \(-1\). Let's take the expression \(-\sqrt{100}\) as an example.
  • First, find the principal square root of the radicand 100, which is 10.
  • Next, multiply 10 by \(-1\) to apply the negative sign, resulting in \(-10\).
This final number, \(-10\), is the evaluated result of \(-\sqrt{100}\). Understanding how and when to apply a negative sign can prevent confusion and ensure accuracy in computations involving square roots.

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Most popular questions from this chapter

For Problems \(19-32\), write each of the following in ordinary decimal notation. $$ (5.123)\left(10^{-8}\right) $$

A square pixel on a computer screen has a side of length (1.17) \(\left(10^{-2}\right)\) inches. Find the approximate area of the pixel in inches. Express the result in decimal form.

The field of view of a microscope is \((4)\left(10^{-4}\right)\) meters. If a single cell organism occupies \(\frac{1}{5}\) of the field of view, find the length of the organism in meters. Express the result in scientific notation.

Convert each number to scientific notation and perform the indicated operations. Express the result in ordinary decimal notation. $$ \frac{360,000,000}{0.0012} $$

(a) Because \(\frac{4}{5}=0.8\), we can evaluate \(10^{\frac{4}{5}}\) by evaluating \(10^{0.8}\), which involves a shorter sequence of "calculator steps." Evaluate parts (b), (c), (d), (e), and (f) of Problem 96 and take advantage of decimal exponents. (b) What problem is created when we try to evaluate \(7^{\frac{4}{3}}\) by changing the exponent to decimal form?
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