Problem 9 - Intermediate Algebra

Question

Graph the following real numbers on a number line. $$ \left\\{8^{2 / 3},(-125)^{1 / 3},-16^{-1 / 4}, 4^{3 / 2},-\left(\frac{9}{100}\right)^{-1 / 2}\right\\} $$

Step-by-Step Solution

Answer

The numbers are 8, 4, -3.33, -0.5, and -5 and should be plotted on a number line in this order.

Step 1: Evaluate the Numbers

Let's evaluate each expression:1. \(8^{2/3}\): The cube root of 8 is 2, and 2 squared is 4, so \(8^{2/3} = 4\).2. \((-125)^{1/3}\): The cube root of -125 is -5, so \((-125)^{1/3} = -5\).3. \(-16^{-1/4}\): The fourth root of 16 is 2, and the inverse is \(1/2\), making it \(-1/2\).4. \(4^{3/2}\): The square root of 4 is 2, and 2 cubed is 8, so \(4^{3/2} = 8\).5. \(-\left(\frac{9}{100}\right)^{-1/2}\): The square root of \(\frac{9}{100}\) is \(\frac{3}{10}\), and the inverse is \(\frac{10}{3}\) making it \(-\frac{10}{3}\) or approximately -3.33.

Step 2: Order the Numbers

List the numbers in decreasing order to prepare for graphing: \(8 = 4^{3/2} > 4 = 8^{2/3} > \frac{10}{3} \approx 3.33 > -1/2 = -16^{-1/4} > -5 = (-125)^{1/3}\).

Step 3: Draw the Number Line

On a horizontal line, place the points for each number in the order determined in Step 2. Mark these positions based on their approximate values:- Near 4, mark \(8^{2/3} = 4\).- Near 8, mark \(4^{3/2} = 8\).- Between -3 and -4 (near -3.33), mark \(-\left(\frac{9}{100}\right)^{-1/2} = -\frac{10}{3}\).- Near -0.5, mark \(-16^{-1/4} = -\frac{1}{2}\).- Exactly on -5, mark \((-125)^{1/3} = -5\).

Step 4: Verify Placement and Label

Re-check each evaluated number’s position in relation to the others to confirm the positions on the number line as they appeared in the ordered set.

Key Concepts

Number Line

The number line is a visual representation of real numbers positioned from left to right. Every point on the number line corresponds to a real number. Numbers increase in value as you move to the right, while they decrease as you move to the left.

When graphing numbers on a number line, you should note the approximate position of each value. Begin by identifying the largest and smallest values in your set to understand the necessary range. Equally spaced marks can help guide where each number falls between others:

• Positive numbers are placed to the right of zero.
• Negative numbers lie to the left of zero.
• Larger positive numbers are further right; likewise, larger negative numbers are further left.

By ordering the numbers and marking their positions clearly, the number line provides an intuitive way to understand their relative sizes.

Rational Exponents

Rational exponents are a way to express roots and powers together using fractions in the exponent. They follow the rule:\[a^{m/n} = \sqrt[n]{a^m}\]This means you take the nth root of a raised to the power of m.

To break this down, each part of the expression has a specific meaning:

• The denominator (n) of the fraction denotes the root to be taken.
• The numerator (m) signifies the power to which the number is raised.

For instance, consider \(8^{2/3}\): you first take the cube root of 8, resulting in 2, and then square 2 to get 4. Rational exponents can therefore represent various operations more succinctly, combining roots and powers in one expression.

Roots and Powers

Roots and powers are fundamental operations in mathematics. A power, represented by an exponent, indicates how many times a number (the base) is multiplied by itself. For example, in \(4^3\), the base 4 is used as the factor 3 times, equaling 64.

Roots, on the other hand, determine what number, when exponentiated to a given power, results in the original number. Commonly used roots include square roots and cube roots:

• The square root of a number \(a\) is a value that, when multiplied by itself, gives \(a\). For instance, the square root of 9 is 3, because \(3^2 = 9\).
• The cube root of a number \(a\) provides the value that, when used as a factor three times, results in \(a\). For example, the cube root of 27 is 3, since \(3^3 = 27\).

Understanding both roots and powers solidifies the ability to manipulate expressions with exponents and solve complex mathematical problems efficiently.