Problem 59 - Algebra and Trigonometry

Question

\(49-68=\) Simplify the expression and eliminate any negative exponent(s). Assume that all letters denote positive numbers. $$ \left(x^{-5} y^{1 / 3}\right)^{-3 / 5} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x^3}{y^{1/5}}\).

Step 1: Simplify the Exponent

The expression given is \((x^{-5} y^{1 / 3})^{-3 / 5}\). To simplify this, use the power of a power property \((a^m)^n = a^{m*n}\). Start by applying the exponent to each factor inside the parenthesis:\[(x^{-5})^{-3/5} \times (y^{1/3})^{-3/5}\]

Step 2: Simplify Each Factor

Now, simplify each part:- For \((x^{-5})^{-3/5}\), multiply the exponents: \(-5 \times -3/5 = 3\), so \(x^3\).- For \((y^{1/3})^{-3/5}\), multiply the exponents: \(1/3 \times -3/5 = -1/5\), so \(y^{-1/5}\).

Step 3: Combine the Simplified Factors

Putting it all together, you get the simplified expression: \[x^3 y^{-1/5}\].

Step 4: Eliminate Negative Exponent

To express the answer without negative exponents, remember that \(a^{-b} = \frac{1}{a^b}\). Therefore, rewrite \(y^{-1/5}\) as \(\frac{1}{y^{1/5}}\). This gives:\[x^3 \times \frac{1}{y^{1/5}} = \frac{x^3}{y^{1/5}}\].

Key Concepts

Negative Exponents

Negative exponents can seem a bit tricky at first, but they have a simple rule behind them. When you see a negative exponent like in the expression \(a^{-n}\), it means that you are dealing with the reciprocal of the base raised to the positive exponent. In simpler terms, \(a^{-n}\) is the same as \(\frac{1}{a^n}\). This means we flip the fraction. For instance, if you have \(x^{-3}\), it can be rewritten as \(\frac{1}{x^3}\). Using this principle, expressions with negative exponents can be rewritten to eliminate the negativity of the exponent.

• Negative exponents indicate division.
• They are a way of expressing reciprocal powers.
• Transform the expression into a positive exponent by reciprocating.

In the original exercise, to eliminate \(y^{-1/5}\), it is converted to \(\frac{1}{y^{1/5}}\). This step is crucial in simplifying expressions.

Power of a Power Property

The power of a power property is a key tool in algebra for simplifying expressions that involve exponents. The rule states that when you raise a power to another power, you multiply the exponents: \((a^m)^n = a^{m \cdot n}\). This simplification process helps break down more complex expressions into something manageable.For example, consider the expression \((x^{-5})^{-3/5}\). According to the power of a power property, you multiply the exponents: \(-5 \times -3/5 = 3\). So, this simplifies to \(x^3\). Similarly, for \((y^{1/3})^{-3/5}\), applying the rule gives \(1/3 \times -3/5 = -1/5\), simplifying to \(y^{-1/5}\).

• Multiply the exponents when using the power of a power rule.
• This property is essential for simplifying nested exponents.

Simplifying Expressions

Simplifying expressions in algebra involves reducing an expression to its simplest form. This often means eliminating negative exponents, combining like terms, and applying fundamental properties of exponents.In the example \( (x^{-5} y^{1/3})^{-3/5} \), the first step is to apply the power of a power property, simplifying inside the parentheses. Once each component is simplified, the next step is to ensure no negative exponents remain in the final answer. This can involve moving parts of a fraction to switch the power from negative to positive, resulting in a cleaner expression.

• Apply exponent rules for simplification.
• Use reciprocal properties to eliminate negative exponents.
• The goal is a clear, simplified expression without negative powers.

In this case, after simplifying \(x^3 y^{-1/5}\) to \(\frac{x^3}{y^{1/5}}\), you have successfully simplified the original expression while eliminating the negative exponent.