Given $$2^2 \times 4^y = \frac{1}{2\sqrt{2}}$$ express $y$ as a function of $x$. - Edexcel - A-Level Maths Pure - Question 3 - 2019 - Paper 2

The term $\frac{1}{2\sqrt{2}}$ can be rewritten as:

$\frac{1}{2 \cdot 2^{1/2}} = \frac{1}{2^{1 + 1/2}} = \frac{1}{2^{3/2}}$

Thus, we have:

$2^{2 + 2y} = 2^{-3/2}$

Since the bases are the same, we can set the exponents equal to each other:

$2 + 2y = -\frac{3}{2}$

To isolate $y$, we first subtract 2 from both sides:

$2y = -\frac{3}{2} - 2$

Converting 2 to a fraction gives:

$2 = \frac{4}{2}$

So, we have:

$2y = -\frac{3}{2} - \frac{4}{2} = -\frac{7}{2}$

Now, divide both sides by 2:

$y = -\frac{7}{4}$