(3m−5n24m−2n0)2⋅(mn49n)2

3 min read Jun 16, 2024
(3m−5n24m−2n0)2⋅(mn49n)2

Simplifying the Expression: (3m−5n24m−2n0)2⋅(mn49n)2

Let's break down the simplification of this expression step by step:

1. Understanding the Expression

The expression involves several multiplications and exponents. It's crucial to remember the order of operations (PEMDAS/BODMAS) to correctly simplify it.

2. Simplifying the Terms within Parentheses

  • (3m−5n24m−2n0)2:
    • 3m - 5n: This term remains as is, as we can't combine the variables.
    • 24m - 2n0: Since any number multiplied by 0 equals 0, this simplifies to 24m.
    • (3m - 5n 24m)2: This represents squaring the entire term. We will expand this in the next step.
  • (mn49n)2:
    • mn: This term remains as is.
    • 49n: This simplifies to 49n, as there is no other term to combine it with.
    • (mn 49n)2: We will expand this in the next step.

3. Expanding the Squares

  • (3m - 5n 24m)2: This expands as (3m - 5n)(3m - 5n)(24m).
  • (mn 49n)2: This expands as (mn)(mn)(49n)(49n).

4. Simplifying the Multiplications

  • (3m - 5n)(3m - 5n)(24m):
    • Multiply the first two terms: (9m² - 15mn - 15mn + 25n²) = (9m² - 30mn + 25n²)
    • Multiply the result by 24m: (9m² - 30mn + 25n²) * 24m = 216m³ - 720m²n + 600mn²
  • (mn)(mn)(49n)(49n):
    • Multiply all the terms: m²n² * 49²n² = 2401m²n⁴

5. Combining the Simplified Terms

Finally, we multiply the simplified terms:

(216m³ - 720m²n + 600mn²) * (2401m²n⁴) = 518616m⁵n⁴ - 1728720m⁴n⁵ + 1440600m³n⁶

Therefore, the simplified form of the expression (3m−5n24m−2n0)2⋅(mn49n)2 is 518616m⁵n⁴ - 1728720m⁴n⁵ + 1440600m³n⁶.

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