a) Write $6^{-2}$ and $8^{rac{1}{2}}$ without using indices - Leaving Cert Mathematics - Question 1 - 2014

From a calculator, we find that $2^{24} = 16,777,216$ which can be expressed as $1.678 \times 10^{7}$.

We can simplify as follows:

Firstly, we write $\sqrt{a} = a^{\frac{1}{2}}$:

$$\frac{(a\sqrt{a})^{3}}{a^{4}} = \frac{(a^{1} a^{\frac{1}{2}})^{3}}{a^{4}} = \frac{(a^{\frac{3}{2}})^{3}}{a^{4}} = \frac{a^{\frac{15}{2}}}{a^{4}} = a^{\frac{15}{2}-4} = a^{\frac{7}{2}} = \sqrt{a}.$$$

We rewrite the equation as follows:

$49^{y} = (7^{2})^{y} = 7^{2y}$

This leads to:

$2y = 2y + 2$

which simplifies to find $y = -1$.

To verify, substituting back gives:

$49^{-1} = 7^{2(-1) + 2}$

which holds true.