Answered

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

$\left(125 x^{-9} y^{-21}\right)^{\frac{4}{3}}$

Verified Answer :

  • Apply the exponent power rule to distribute \(\frac{4}{3}\) to each factor: \(125^{\frac{4}{3}} \cdot (x^{-9})^{\frac{4}{3}} \cdot (y^{-21})^{\frac{4}{3}}\).
  • Simplify each term individually: \(625\), \(x^{-12}\), and \(y^{-28}\).
  • Convert the negative exponents into positive exponents in the denominator.
  • The final simplified expression is \(\boxed{\frac{625}{x^{12} y^{28}}}\).

Explanation

  1. Analyzing the Expression

The expression to simplify is \(\left(125 x^{-9} y^{-21}\right)^{\frac{4}{3}}\). We have three main components inside the parentheses: a constant 125 and two variables \(x\) and \(y\) with negative exponents. To simplify this, we need to distribute the power \(\frac{4}{3}\) to each of these components, apply the exponentiation rule \((a^m)^n = a^{m \cdot n}\), and convert negative exponents into positive ones using \(x^{-n} = \frac{1}{x^n}\).

  1. Applying Exponent Rules

Distribute the exponent \(\frac{4}{3}\) to each factor inside the parentheses:
\[\left(125 x^{-9} y^{-21}\right)^{\frac{4}{3}} = 125^{\frac{4}{3}} \cdot (x^{-9})^{\frac{4}{3}} \cdot (y^{-21})^{\frac{4}{3}}\]
Now, calculate each part:

  • For the constant: \(125^{\frac{4}{3}} = (5^3)^{\frac{4}{3}} = 5^{3 \cdot \frac{4}{3}} = 5^4 = 625\).
  • For \(x\): \((x^{-9})^{\frac{4}{3}} = x^{-9 \cdot \frac{4}{3}} = x^{-3 \cdot 4} = x^{-12}\).
  • For \(y\): \((y^{-21})^{\frac{4}{3}} = y^{-21 \cdot \frac{4}{3}} = y^{-7 \cdot 4} = y^{-28}\).
  1. Final Simplification

Combine the results from the previous step:
\[625 \cdot x^{-12} \cdot y^{-28}\]
To write the expression using only positive exponents, apply the rule \(a^{-n} = \frac{1}{a^n}\) to the variables \(x^{-12}\) and \(y^{-28}\):
\[625 \cdot \frac{1}{x^{12}} \cdot \frac{1}{y^{28}} = \frac{625}{x^{12} y^{28}}\]
The final simplified expression, written with positive exponents, is \(\boxed{\frac{625}{x^{12} y^{28}}}\).

Examples

In physics, particularly when working with scaling laws or dimensional analysis, expressions like this are common. For example, if the power output \(P\) of a certain device scales with its dimensions \(x\) and \(y\) according to the given formula, simplifying the expression helps engineers quickly determine how changes in dimensions \(x\) and \(y\) affect the final power output, making it easier to optimize the device's design.

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