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Chapter 1: Problem 13

Graph each number on a number line. $$ -\sqrt{24} $$

Short Answer

Expert verified
The number \( -\sqrt{24} \) is approximately equal to \( -4.9 \) and is plotted just to the left of \( -5 \) on the number line.

Step by step solution

01

Simplify the Radical

Before plotting the number on a number line, simplify \( -\sqrt{24} \) by factoring out squares from the radicand, 24. The number 24 can be factored into \( 4 \times 6 \) where 4 is a perfect square. Thus, \( -\sqrt{24} = -\sqrt{4 \times 6} = -\sqrt{4} \cdot \sqrt{6} = -2\sqrt{6} \) since \( \sqrt{4} = 2 \) and approximately, \( \sqrt{6} \approx 2.45 \) hence \( -2\sqrt{6} \approx -4.9 \) approximately.
02

Plot the Number on the Number Line

Now that we know that \( -\sqrt{24} \) is approximately \( -4.9 \) we will locate this value on the number line. Start at zero, move left since it is negative, and approximate the position between \( -4 \) and \( -5 \) where \( -4.9 \) would be just slightly to the left of \( -5 \) as \( -4.9 \) is just short of \( -5 \) on the number line.

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Key Concepts

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

Simplifying Radicals
Understanding how to simplify radicals is essential for accurately representing non-integer numbers and managing complex calculations. Simplifying a radical involves breaking down the number under the radical sign, known as the radicand, into its factors, ideally finding perfect squares which make the radical easier to compute.

To simplify \( -\sqrt{24} \), we look for the largest perfect square that divides into 24, which is 4. The radical then becomes \( -\sqrt{4 \times 6} \) or \( -2\sqrt{6} \), since \( \sqrt{4} \) is 2. This process is vital because it simplifies the expression to a more manageable form, making further operations or graphing on a number line possible.
Number Line
A number line is a visual representation of numbers in a horizontal array, where each point corresponds to a number. Positive numbers are typically plotted to the right of zero, and negative numbers to the left. It's an intuitive way to understand the ordering and relative size of numbers.

When graphing numbers, especially variables or radicals, number lines are incredibly helpful. They allow students to visualize the distance and relation between numbers, which is especially valuable when dealing with negative values and complex expressions.
Radical Expressions
Radical expressions, like \( \sqrt{n} \), can often seem daunting. They represent the root of a given number or expression which can be whole numbers, fractions, variables, or a combination thereof. When radicals include square roots, they symbolize the principal square root, or the non-negative square root of a number.

To understand and manipulate these expressions confidently, students should familiarize themselves with properties of radicals and methods of simplification. This foundation makes working with more complex algebraic expressions that contain radicals far more approachable.
Plotting Negative Numbers
Graphing negative numbers may initially intimidate some students, but it follows a simple logic akin to plotting positive numbers, albeit in the opposite direction on a number line. Starting at zero, any negative number is found by moving leftward. The farther from zero, the smaller the number.

For instance, \( -4.9 \) would be just a hair's breadth to the left of \( -5 \) on the number line because it's slightly greater (or less negative). Visualizing fractions and decimals between the whole numbers is a valuable skill, helping to accurately graph and understand numbers, which are not whole, such as those that result from simplifying radicals.

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Most popular questions from this chapter

Simplify by combining like terms. $$ w+3 z+5 w+2 z $$

The sum of the lengths of any two sides of a triangle is greater than the length of the third side. In \(\triangle A B C, B C=4\) and \(A C=8-A B\) . What can you conclude about \(A B ?\)

Which expression has the greatest value for \(x=-4\) and \(y=5 ?\) $$ \begin{array}{llll}{\text { A. }-3 x y} & {\text { B. } 3 x y} & {\text { C. } x y^{2}} & {\text { D. } x^{2} y}\end{array} $$

The random number table simulates an experiment where you toss a coin 90 times. Even digits represent heads and odd digits represent tails. What is the experimental probability, to the nearest percent, of the coin coming up heads? Random Number Table $$\begin{array}{lll}{31504} & {51648} & {40613} \\ {79321} & {80927} & {42404} \\ {15594} & {84675} & {68591} \\ {34178} & {00460} & {31754} \\\ {49676} & {58733} & {00884} \\ {85400} & {72294} & {22551}\end{array}$$ A. 45\(\%\) B. 50\(\%\) C. 54\(\%\) D. 56\(\%\)

Solve each equation. Check for extraneous solutions. $$ |2 x+5|=3 x+4 $$
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