Question

\(\left(3^{-2 / 3} \cdot 3^{1 / 3}\right)^{-1}\)

Short answer

The simplified form of the given expression is \(3^{1 / 3}\)

Step-by-step solution

Simplify the Inner Expression

Begin with simplifying the inner expression: \(\left(3^{-2 / 3} \cdot 3^{1 / 3}\right)\). When multiplying terms with the same base, you add the exponents. So this can simplified to: \(3^{-2 / 3 + 1 / 3} = 3^{-1 / 3}\)

Simplify the Outer Expression

Then address the outer negative exponent, that is \( \left(3^{-1 / 3}\right)^{-1}\), using the rule that \(a^{-n} = 1/a^n\), equals to: \(1/(3^{-1 / 3}) = 3^{1 / 3}\)

Conclusion

So, the simplification of the given expression \(\left(3^{-2 / 3} \cdot 3^{1 / 3}\right)^{-1}\) results in \(3^{1 / 3}\)

Key concepts

Simplifying Expressions

The process of simplifying expressions is akin to tidying up a messy room by organizing items and putting them in their right places. In algebra, simplification involves combining like terms, performing arithmetic, and reducing expressions to their most basic form without changing their value. When simplifying expressions, one can often use exponent rules to combine terms with the same base.

For example, consider the expression
\[\begin{equation}\left(3^{-2 / 3} \cdot 3^{1 / 3}\right)^{-1}\end{equation}\]
This looks a little complex at first, but by understanding the properties of exponents, it can be greatly simplified. The key is to combine exponents when bases are the same, which leads to addition or subtraction of the exponent values. For the inner expression, you have two powers of 3 that can be combined, simplifying the expression considerably.

Negative Exponents

Imagine you have a debt that cancels out a portion of your savings. Negative exponents work a bit like this in mathematical terms. They indicate a reciprocal, or an inverse. For any non-zero number 'a' and positive integer 'n', the rule
\[\begin{equation}a^{-n} = \frac{1}{a^n}\end{equation}\]
applies, saying that a number raised to a negative exponent is equal to one divided by that number raised to the opposite positive exponent.

When working with the expression from our example, you deal with a power to a negative exponent. To simplify this, you must recognize that to 'resolve' a negative exponent, you can rewrite it as the reciprocal of the base raised to the corresponding positive exponent. This rule is the turning point to understanding how to simplify expressions that include negative exponents, leading you much closer to the simplest form of the expression.

Properties of Exponents

Expanding your toolkit for algebra includes mastering the properties of exponents. These properties are the rules that guide the manipulation of exponential expressions. Key properties include:

• The Product Rule: To multiply with the same base, add the exponents: \[\begin{equation}a^m \cdot a^n = a^{m+n}\end{equation}\]
• The Quotient Rule: To divide with the same base, subtract the exponents: \[\begin{equation}a^m / a^n = a^{m-n}\end{equation}\]
• The Power Rule: To raise a power to another power, multiply the exponents: \[\begin{equation}(a^m)^n = a^{m \cdot n}\end{equation}\]
• The Negative Exponent Rule: A negative exponent represents a reciprocal: \[\begin{equation}a^{-n} = 1/a^n\end{equation}\]

Understanding and applying these properties allows for efficient simplification of algebraic expressions, much like using proper tools makes a job easier. In our example, both the product rule and the power rule are utilized to arrive at the simplified solution.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operations (like addition, subtraction, multiplication, and division). Think of it as a sentence where numbers and operations tell a story, and the variables are the characters.

The expression we've been unraveling represents a challenging sentence but broken down into manageable parts using the tools provided by properties of exponents, it can be understood and simplified. The final step in our problem takes our simplified inner expression and further refines it by applying the negative exponent rule to arrive at a simple and elegant algebraic expression:
\[\begin{equation}3^{1 / 3}\end{equation}\]
Algebraic expressions often require several steps of simplification, just as complex sentences need to be read and understood piece by piece. As you practice, these steps become second nature and your mathematical literacy improves, turning seemingly complicated expressions into solvable puzzles.