Question

Simplify each expression. Express the answer so that all exponents are positive. Whenever an exponent is 0 or negative, we assume that the base is not \(0 .\) \(\left(x^{-1} y\right)^{3}\)

Short answer

\( \frac{y^{3}}{x^{3}} \)

Step-by-step solution

Apply the power to each term inside the parenthesis

Raise each term inside the parenthesis to the power of 3: ewline ewline \(\left(x^{-1} y\right)^{3} = (x^{-1})^{3} \cdot y^{3}\)

Simplify the expression

Apply the exponent rule \(\left( a^m \right)^n = a^{m \cdot n}\): ewline ewline \( (x^{-1})^{3} \cdot y^{3} = x^{-1 \cdot 3} \cdot y^{3} = x^{-3} \cdot y^{3} \)

Express exponents as positive

To express negative exponents as positive, use the rule \(a^{-n} = \frac{1}{a^n}\): ewline ewline \( x^{-3} \cdot y^{3} = \frac{1}{x^{3}} \cdot y^{3}\)

Write the final answer

Combine the terms: ewline ewline \( \frac{y^{3}}{x^{3}}\)

Key concepts

Negative Exponents

Understanding negative exponents is essential for simplifying expressions. A negative exponent indicates that the base should be moved to the denominator of a fraction and then raised to the corresponding positive exponent. For example, if we have an expression like \(x^{-3}\), we can rewrite it as \(\frac{1}{x^{3}}\). This simplifies the process of dealing with negative exponents and helps in converting them into positive exponents which are often easier to manage. Remember, keeping the base the same is important, only the exponent changes its sign and position.

Power Rule

The power rule is a basic principle in algebra that deals with exponents. It states that \((a^m)^n = a^{m \cdot n}\). This means when you raise a power to another power, you multiply the exponents. For instance, in our exercise \((x^{-1})^3\), we apply the power rule to get \((x^{-1})^3 = x^{-1 \cdot 3} = x^{-3}\). Similarly, \(y^3\) remains unchanged. The power rule is extremely useful for simplifying expressions and making calculations more manageable.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and using algebraic rules to make the expression as simple as possible. It usually involves several steps:

• Apply any relevant algebraic rules, like the power rule.
Fill in the brackets and combine like terms.
• Ensure all exponents are expressed as positive by using the negative exponent rule.

In our example, we originally had \((x^{-1} y)^3\). Using the power rule and rules for negative exponents, we turned this into \(\frac{y^{3}}{x^{3}}\), which is a simplified expression with all positive exponents.

Positive Exponents

Working with positive exponents is usually more straightforward and preferred because they represent standard multiplication. Once we've converted any negative exponents into positive ones, our expressions become easier to understand and solve. For the problem discussed, we ultimately converted \((x^{-1})^3\) into \(\frac{1}{x^{3}}\), resulting in the final simplified expression \(\frac{y^{3}}{x^{3}}\). Always check for positive exponents when presenting your final answer to make sure it’s in a standardized form, ready for any further operations if needed.