Verified Answer :

  • The expression consists of two terms with the same variable part \(x^5\).
  • Using the distributive property, we add the coefficients: \(3 + 4 = 7\).
  • The simplified expression is \(7x^5\).
  • The missing coefficient is \(\boxed{7}\).

Explanation

  1. Analyzing the expression

To simplify the expression \(3x^5 + 4x^5\), we observe that both terms contain the same variable part, \(x^5\). Expressions of the form \(ax^n + bx^n\) can be simplified using the distributive property of multiplication over addition, which states that \(ax^n + bx^n = (a+b)x^n\). In our case, \(a = 3\), \(b = 4\), and \(n = 5\).

  1. Combining like terms

Using the distributive property, we combine the coefficients by adding them together:
\[3x^5 + 4x^5 = (3 + 4)x^5\]
Calculating the sum of the coefficients, we find:
\[3 + 4 = 7\]
Therefore, the expression becomes:
\[7x^5\]

  1. Final Conclusion

The expression \(3x^5 + 4x^5\) simplifies to \(7x^5\). Thus, the missing coefficient is \(\boxed{7}\).

Examples

In real-world scenarios, combining like terms is essential for simplifying calculations in finance or engineering. For example, if you are calculating the total cost of two different bulk orders of components, each costing some amount \(P\) per unit raised to a specific power (like scaling factors in logistics), you can group them to find the total multiplier. If Order A requires \(3x^5\) units and Order B requires \(4x^5\) units, the total is simply \(7x^5\) units. This makes managing inventory and budgeting much more efficient.

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