Problem 37 - Algebra A Combined Function

Question

Write with positive exponents. Simplify if possible. $$ \frac{1}{a^{-2 / 3}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( a^{2/3} \) with positive exponents.

Step 1: Understand Negative Exponents

Recall that a negative exponent indicates that the base is on the wrong side of a fraction. For example, \( x^{-n} = \frac{1}{x^n} \) and \( \frac{1}{x^{-n}} = x^n \).

Step 2: Apply the Negative Exponent Rule

Given the expression \( \frac{1}{a^{-2/3}} \), we apply the rule that \( \frac{1}{x^{-n}} = x^n \). By doing this, \( \frac{1}{a^{-2/3}} \) becomes \( a^{2/3} \).

Step 3: Write with Positive Exponents

The expression is now \( a^{2/3} \), which is already expressed with positive exponents.

Key Concepts

Positive Exponents

Understanding positive exponents is crucial when working with expressions. When an exponent is positive, it means the base is being multiplied by itself a certain number of times. For instance, with the expression \( a^n \), \( a \) is the base and \( n \) is the exponent, indicating that \( a \) should be multiplied by itself \( n \) times. This helps simplify expressions and makes calculations more straightforward.

Positive exponents reflect direct multiplication rather than division or inversion, as negative exponents often do. Utilizing positive exponents helps prevent misunderstandings and keeps expressions as simple as possible.

• For \( a^2 \), you multiply \( a \times a \).
• For \( a^3 \), you multiply \( a \times a \times a \).

Simplifying Expressions

Simplifying expressions is a vital skill in algebra that involves reducing expressions to their simplest form. The primary goal is to make the expression as easy to work with as possible.

To simplify an expression, follow these general guidelines:

• Combine like terms, which are terms that have the same variables raised to the same power.
• Apply exponent rules, such as converting negative exponents to positive by relocating the base within the fraction.
• Factor where possible to further condense the expression.

A simplified expression is neat and efficient, and it often aids in understanding the problem better or in solving equations more easily. By converting \( \frac{1}{a^{-2/3}} \) to \( a^{2/3} \), the expression becomes cleaner and eliminates any potential confusion related to negative exponents.

Fractional Exponents

Fractional exponents are another critical concept in algebra, representing roots of numbers. They are similar to the standard exponents, but they provide more versatility in calculations. A fractional exponent, such as \( a^{m/n} \), signifies the \( n \)-th root of \( a \) raised to the \( m \)-th power.

Breaking this into steps:

• The denominator \( n \) of the fraction indicates the root. For example, \( a^{1/3} \) represents the cube root of \( a \).
• The numerator \( m \) shows the power applied to the result of the root. Therefore, \( a^{2/3} \) represents the cube root of \( a \), which is then squared.

Fractional exponents offer a compact and efficient way to work with roots and powers, enhancing both simplicity and clarity when dealing with complex expressions.