Simplifying indices

Question

Probably a straightforward question but I have a mind blank.

I'm back into a maths course after 20 years and am working on algebra but I cannot see the workings of this answer and it's bugging me.

I am dividing a large set of indices and have to express the answer Innindice form. The denominator of the sum is 9 to the power 2 X 8 to the power 3.

The answer is to simplify it to 3 to the power 5 X 2 to the power 9. Is there an actual method to this simplification? Or is it just an understanding that 9 divided by 3 therefor there must be a simpler expression? Arrrrggghhhh

Answer 1

Use the rule that $\displaystyle (a^b)^c = a^{bc}$. So $9^2$ is the same thing as $\displaystyle (3^2)^2 = 3^{2 \times 2} = 3^4$.

Same for $8$. Note that $8 = 2^3$ so $\displaystyle 8^3 = (2^3)^3 = 2^{3 \times 3} = 2^9$.

This follows for integers because $$(a^b)^c = \underbrace{a^{b} \times a^b \times \cdots a^b}_{c \, \text{times}} = a^{b + b + b+ \cdots b} = a^{bc}$$

Answer 2

$$9^2 \times 8 ^3 = (3^2)^2 \times (2^3)^3= 3^{2 \times 2} \times 2^{3\times 3}=3^4 \times 2^9$$