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

Question 20

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

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

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

Step 1

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Answer

We first express all terms using base 3. Recall that:

Thus, we can rewrite the equation:

$\left(3^2\right)^{\frac{1}{2}} = \left(3^3\right)^{x} + 3^{1+x}$

This simplifies to:

$3^{1} = 3^{3x} + 3^{1+x}$

Step 2

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Answer

Now, we factor out the common term from the right side:

$3^{1} = 3^{1+x}(1 + 3^{2x - (1+x)})$

This means we have:

$3^1 = 3^{1+x}$

Equating the exponents gives:

$1 = 1 + x$

Step 3

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101 rated

Answer

From the equation $1 = 1 + x$ , we can solve for x:

$x = 0$

Thus, the exact value of $x$ is:

$x = 0$

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