Answered

Simplify the following expression to simplest form using only positive exponents.

$\left(32 x^5 y^{-35}\right)^{-\frac{3}{5}}$

Verified Answer :

  • Distribute the exponent \(-\frac{3}{5}\) to each factor inside the parenthesis: \(32^{-\frac{3}{5}} x^{-3} y^{21}\).
  • Simplify the constant \(32^{-\frac{3}{5}}\) to \(\frac{1}{8}\).
  • Rewrite the variable \(x^{-3}\) as \(\frac{1}{x^3}\) to ensure only positive exponents.
  • Combine the results to get \(\boxed{\frac{y^{21}}{8x^3}}\).

Explanation

  1. Problem Analysis and Strategy

We are given the expression \(\left(32 x^5 y^{-35}\right)^{-\frac{3}{5}}\) and aim to simplify it using only positive exponents. We will use the properties of exponents to break down the expression and simplify each part step-by-step. The key rules we will apply are:

  1. Power of a product: \((abc)^n = a^n b^n c^n\)
  2. Power of a power: \((a^m)^n = a^{m \cdot n}\)
  3. Negative exponent rule: \(a^{-n} = \frac{1}{a^n}\)
  1. Distributing the Exponent

First, we distribute the exponent \(-\frac{3}{5}\) to each factor within the parentheses using the power of a product rule:
\[\left(32 x^5 y^{-35}\right)^{-\frac{3}{5}} = 32^{-\frac{3}{5}} \cdot (x^5)^{-\frac{3}{5}} \cdot (y^{-35})^{-\frac{3}{5}}\]
Now, we simplify each of these factors individually.

  1. Simplifying Individual Factors
  • For the constant term: Note that \(32 = 2^5\). Thus,

\[32^{-\frac{3}{5}} = (2^5)^{-\frac{3}{5}} = 2^{5 \cdot (-\frac{3}{5})} = 2^{-3} = \frac{1}{2^3} = \frac{1}{8}.\]

  • For the \(x\) variable:

\[(x^5)^{-\frac{3}{5}} = x^{5 \cdot (-\frac{3}{5})} = x^{-3} = \frac{1}{x^3}.\]

  • For the \(y\) variable:

\[(y^{-35})^{-\frac{3}{5}} = y^{-35 \cdot (-\frac{3}{5})} = y^{7 \cdot 3} = y^{21}.\]

  1. Combining and Finalizing

Finally, we combine the simplified factors into a single expression:
\[\frac{1}{8} \cdot \frac{1}{x^3} \cdot y^{21} = \frac{y^{21}}{8x^3}.\]
This expression now contains only positive exponents, which satisfies the problem requirements.

Examples

In scientific and engineering calculations, expressions with negative or fractional exponents are common when dealing with scaling laws, such as how the volume of an object scales with its length. By simplifying such complex expressions, one can quickly determine how changing one variable (like length or density) affects the overall result, making it easier to analyze physical phenomena or perform computations accurately.

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