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(9^x-3^x+1)^2+8*3^x+1 8*9^x+20

2 min read Jun 16, 2024
(9^x-3^x+1)^2+8*3^x+1 8*9^x+20

Simplifying the Expression: (9^x-3^x+1)^2+83^x+1 89^x+20

This expression can be simplified using algebraic manipulation and understanding the properties of exponents. Let's break it down step-by-step:

1. Expanding the Square

  • (9^x-3^x+1)^2: This is a squared binomial. We can expand it using the formula (a-b+c)^2 = a^2 + b^2 + c^2 - 2ab + 2ac - 2bc.

    • (9^x)^2 = 81^x
    • (3^x)^2 = 9^x
    • (1)^2 = 1
    • -2*(9^x)(3^x) = -2(27^x)
    • 2*(9^x)(1) = 2(9^x)
    • -2*(3^x)(1) = -2(3^x)

    So, (9^x-3^x+1)^2 = 81^x + 9^x + 1 - 2(27^x) + 2(9^x) - 2*(3^x)**

2. Rearranging Terms

  • Let's combine similar terms:
    • 81^x + 9^x + 1 - 2(27^x) + 2(9^x) - 2*(3^x) + 8*(3^x) + 8*(9^x) + 20**
    • 81^x - 2(27^x) + 10(9^x) + 6*(3^x) + 21**

3. Factoring (Optional)

  • Although not essential, we can factor out common terms:

    • (9^x - 2)(81^x + 10) + 6(3^x) + 21

Final Simplified Expression:

The simplified expression is 81^x - 2(27^x) + 10(9^x) + 6*(3^x) + 21** or alternatively, (9^x - 2)(81^x + 10) + 6(3^x) + 21.

Note: This expression can be further manipulated based on specific requirements or desired format.

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