Problem 66 - Algebra 1: Concepts and Skills

Question

Simplify the expression \(\left(\frac{2}{9}\right)^{-3}\) F. \(-\frac{6}{27}\) G. \(\frac{27}{6}\) H. \(-\frac{8}{729}\) J. \(\frac{729}{8}\)

Step-by-Step Solution

Answer

J. \(\frac{729}{8}\)

Step 1: Understand the negative exponent property

Recall the negative exponent rule: \(a^{-n} = \frac{1}{a^{n}}\). Using the rule, turn the negative exponent into a positive exponent by finding the reciprocal of the base.

Step 2: Apply negative exponent property

Applying the negative exponent rule to \(\left(\frac{2}{9}\right)^{-3}\) gives \(\left(\frac{9}{2}\right)^{3}\)

Step 3: Simplification.

Compute \(\left(\frac{9}{2}\right)^{3}\) which simplifies to cube of 9 (729) divided by cube of 2 (8). Hence, the simplified expression is \(\frac{729}{8}\)

Key Concepts

Reciprocal

In mathematics, understanding the concept of a reciprocal is essential, especially when working with exponents. A reciprocal is simply the inverse of a number or fraction. For any non-zero number or fraction \(a\), its reciprocal is \(\frac{1}{a}\). This is the same for fractions, where you flip the numerator and the denominator to find its reciprocal.
The process of finding the reciprocal is especially useful when dealing with negative exponents. When you encounter a negative exponent, the first step is to take the reciprocal of the base of the expression. For example, in the expression \(\left(\frac{2}{9}\right)^{-3}\), the reciprocal of \(\frac{2}{9}\) is \(\frac{9}{2}\). By flipping the fraction, you change the sign of the exponent to positive, preparing the expression for further operations.

Exponentiation

Exponentiation is a fundamental mathematical operation involving a base and an exponent. When a base is raised to an exponent, it is multiplied by itself a number of times equal to the exponent. For example, when calculating \(\left(\frac{9}{2}\right)^3\), we multiply \(\frac{9}{2}\) by itself three times: \(\left(\frac{9}{2}\right) \times \left(\frac{9}{2}\right) \times \left(\frac{9}{2}\right)\).
This operation is often used for handling expressions with exponents, both positive and negative. After converting a negative exponent to a positive one using the reciprocal, you can compute the new expression using standard exponentiation rules.

• Positive exponents indicate repeated multiplication.
• Negative exponents indicate the need for reciprocation.

Understanding these basics helps simplify complex expressions involving exponents.

Simplification

Simplification is the process of reducing expressions to their simplest form. In this exercise, after converting the negative exponent into a positive one, the next step is to simplify \(\left(\frac{9}{2}\right)^3\). This means computing both the numerator and the denominator separately.
To simplify:

• Cube the numerator: \(9^3 = 729\).
• Cube the denominator: \(2^3 = 8\).

The expression simplifies to \(\frac{729}{8}\). Simplification can reveal the precise value or structure of the expression. This step is crucial in ensuring calculations are correct and they often lead to more manageable numbers or expressions. Always check each component carefully for accurate simplification.