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Given that $$9^{ rac{1}{2}} = 27^{x} + 3^{x}$$ find the exact value of x. - Edexcel - GCSE Maths - Question 19 - 2019 - Paper 1

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Given-that--$$9^{-rac{1}{2}}-=-27^{x}-+-3^{x}$$-find-the-exact-value-of-x.-Edexcel-GCSE Maths-Question 19-2019-Paper 1.png

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

Worked Solution & Example Answer:Given that $$9^{ rac{1}{2}} = 27^{x} + 3^{x}$$ find the exact value of x. - Edexcel - GCSE Maths - Question 19 - 2019 - Paper 1

Step 1

Step 1: Convert to a common base

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Answer

We can start by expressing all terms on the left-hand side in terms of powers of 3. Since 9=329 = 3^2, we have: 9^{ rac{1}{2}} = (3^2)^{ rac{1}{2}} = 3^{1} = 3. Thus, the equation simplifies to: 3=27x+3x3 = 27^{x} + 3^{x}.

Step 2

Step 2: Simplify the terms

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Answer

Next, we express 27 as a power of 3: 27=3327 = 3^3, therefore: 27x=(33)x=33x27^{x} = (3^3)^{x} = 3^{3x}. Substituting this back into the equation gives: 3=33x+3x3 = 3^{3x} + 3^{x}.

Step 3

Step 3: Solve for x

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Answer

Now we can rewrite the equation: 3=33x+3x3 = 3^{3x} + 3^{x} We can express 3 as 313^1, leading to: 31=33x+3x3^1 = 3^{3x} + 3^{x}. This implies: 31=3x(32x+1)3^{1} = 3^{x}(3^{2x} + 1) Setting the bases equal gives: 1=32x+11 = 3^{2x} + 1, which simplifies to: 32x=23^{2x} = 2. Taking logarithm on both sides:

\x = rac{1}{2} ext{ log}_3(2).$$

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