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Given that $9^{- rac{1}{2}} = 27^{x} + 3^{1+x}$ find the exact value of x. - Edexcel - GCSE Maths - Question 20 - 2019 - Paper 1

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Given-that-$9^{--rac{1}{2}}-=-27^{x}-+-3^{1+x}$-find-the-exact-value-of-x.-Edexcel-GCSE Maths-Question 20-2019-Paper 1.png

Given that $9^{- rac{1}{2}} = 27^{x} + 3^{1+x}$ find the exact value of x.

Worked Solution & Example Answer:Given that $9^{- rac{1}{2}} = 27^{x} + 3^{1+x}$ find the exact value of x. - Edexcel - GCSE Maths - Question 20 - 2019 - Paper 1

Step 1

Convert to a Common Base

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Answer

First, we can express the bases as powers of 3:

  • 99 can be written as 323^2, so:
    9^{- rac{1}{2}} = (3^2)^{- rac{1}{2}} = 3^{-1}

  • 2727 can be written as 333^3, so:
    27x=(33)x=33x27^{x} = (3^3)^{x} = 3^{3x}

Now, substituting these into the equation gives us:
3−1=33x+31+x3^{-1} = 3^{3x} + 3^{1+x}

Step 2

Combine the Terms

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Answer

Next, since the bases are the same, we can combine and equate the exponents:

3−1=33x+31+x3^{-1} = 3^{3x} + 3^{1+x}

This leads us to isolate the terms:
- rac{1}{2} = 3x + 1 + x

Where 1+x1 + x comes from the addition of the powers.

Step 3

Solve for x

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Answer

Now, let's simplify the equation:

- rac{1}{2} = 4x + 1

Subtracting 1 from both sides:

- rac{1}{2} - 1 = 4x
- rac{3}{2} = 4x

Finally, dividing both sides by 4 to solve for x:

x = - rac{3}{8}

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