Verified Answer :

  • Separate the coefficients and variable powers: \(\frac{4}{2} \cdot \frac{x^7}{x^4} \cdot \frac{y^5}{y^3}\).
  • Simplify the coefficient \(\frac{4}{2}\) to \(2\).
  • Apply the quotient rule for exponents (\(a^m / a^n = a^{m-n}\)) to \(x\) and \(y\), obtaining \(x^3\) and \(y^2\).
  • The simplified expression is \(\boxed{2x^3y^2}\).

Explanation

  1. Decomposing the Expression

To simplify the expression \(\frac{4x^7y^5}{2x^4y^3}\), we can break the expression down into its constituent parts: the coefficients and the variable powers. The given expression is:
\[\frac{4x^7y^5}{2x^4y^3} = \left(\frac{4}{2}\right) \times \left(\frac{x^7}{x^4}\right) \times \left(\frac{y^5}{y^3}\right).\]
This allows us to handle each part independently.

  1. Simplifying Coefficients and Exponents

First, we simplify the coefficients:
\[\frac{4}{2} = 2.\]
Next, we use the quotient rule for exponents, which states that for any non-zero base \(a\), \(\frac{a^m}{a^n} = a^{m-n}\). Applying this to the variable terms:
For \(x\): \(\frac{x^7}{x^4} = x^{7-4} = x^3.\)
For \(y\): \(\frac{y^5}{y^3} = y^{5-3} = y^2.\)
Combining these, we get \(2 \times x^3 \times y^2 = 2x^3y^2.\)

  1. Final Result

By combining the results from the previous steps, the expression simplifies to:
\[\boxed{2x^3y^2}.\]

Examples

Algebraic simplification is essential in fields like physics and engineering, where complex formulas are derived to describe natural phenomena. For instance, if you are modeling the volume of a geometric shape where dimensions are given by power functions, you might end up with complex algebraic expressions. Simplifying these terms allows you to see the underlying relationship between variables, making it easier to calculate values and predict behavior in real-world scenarios.

Related Questions