Problem 20 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without radicals or exponents. $$ -\left(-\frac{8}{27}\right)^{-2 / 3} $$

Step-by-Step Solution

Answer

The simplified number with no radicals or exponents is \(-\frac{4}{9}\).

Step 1: Rewrite the negative exponent

Rewrite the negative exponent as a positive exponent by flipping the fraction. $$ -\left(-\frac{8}{27}\right)^{-2 / 3} = -\left(\frac{-8}{27}\right)^{2 / 3} $$

Step 2: Convert the exponent into a radical

Now, let's rewrite the expression with exponent as a radical. $$ -\left(\frac{-8}{27}\right)^{2 / 3} = -\sqrt[3]{\left(\frac{-8}{27}\right)^{2}} $$ Here, the cube root denotes the index 3 in the denominator of the exponent.

Step 3: Apply the exponent rule

We will raise the entire fraction inside the parentheses to the power of 2. $$ -\sqrt[3]{\left(\frac{-8}{27}\right)^{2}} = -\sqrt[3]{\frac{(-8)^{2}}{(27)^{2}}} $$

Step 4: Evaluate the powers

Evaluate the power of 2 for both the numerator and the denominator. $$ -\sqrt[3]{\frac{(-8)^{2}}{(27)^{2}}} = -\sqrt[3]{\frac{64}{729}} $$

Step 5: Evaluate the cube root

Now, evaluate the cube root in the expression. $$ -\sqrt[3]{\frac{64}{729}} = -\frac{4}{9} $$ Therefore, the simplified number with no radicals or exponents is \(-\frac{4}{9}\).

Key Concepts

Negative Exponents

Understanding negative exponents is essential to simplifying expressions and can transform how we interpret equations. To manage a negative exponent, we essentially take the reciprocal of the base and then apply the positive exponent. In our exercise, the negative exponent \( -2/3 \) flips the base fraction, resulting in \( (-8/27)^{2/3} \).

This means that, instead of multiplying the base by itself for a negative number of times (which is counterintuitive), we're doing the opposite - we're repeatedly dividing the base. Remember, \( a^{-n} = 1/a^n \), which neatly captures the essence of working with negative exponents.

Radical Expressions

Radical expressions encompass numbers or expressions under the root symbol. The cube root is a type of radical where we search for a number that, when multiplied by itself three times, gives the original number. In our example, \( \sqrt[3]{(-8/27)^{2}} \) is a radical expression that involves finding the cube root of a fraction squared.

We sometimes use radical notation interchangeably with fractional exponents because they convey the same operation. The index of the radical (3 in the case of a cube root) corresponds to the denominator in the fractional exponent. Simplifying radical expressions often involves transforming them into exponent form, making application of exponent rules easier.

Exponent Rules

There are several exponent rules that make dealing with powers straightforward. For instance, \( (a/b)^n = a^n / b^n \), allows us to separately raise the numerator and the denominator of a fraction to the power of \( n \). Additionally, \( (ab)^n = a^n b^n \) demonstrates that a product raised to a power means raising each factor to that power.

In our exercise, we applied an exponent rule in step 3 where \( (-8)^{2} \) and \( (27)^{2} \) were calculated individually, simplifying inside the cube root. This rule helps break down complex expressions into simpler components that are more easily evaluated.

Cube Root

The cube root is a special operation that asks the question: 'What number multiplied by itself three times equals the given number?' The cube root of a number, say \( x \), is written as \( \sqrt[3]{x} \). It's important to recognize that unlike square roots, cube roots can also apply to negative numbers. For example, \( \sqrt[3]{-8} = -2 \) because \( (-2) * (-2) * (-2) = -8 \).

In the final step of our calculation, we found the cube root of 64/729 to be 4/9 since \( 4 * 4 * 4 = 64 \) and \( 9 * 9 * 9 = 729 \). The cube root is an essential concept in algebra and extends into geometric interpretations, linking the operation to the volume of cubes and three-dimensional analysis.