Can the product of two expressions(exponents of different base and power) be reduced into one expression of one base and power?

Question

Suppose you have the product of two expressions: $ 2^5$ * $ 2^2$, the result will be : $ 2^{5+2}$ = $ 2^7$. This is because we know the exponent rule that if they have the same base, we can add the power. Is there a way to express the product of two expressions of different bases and powers into 1 expression with one base and power, like: $ 3^7$ * $ 4^8$ = $ a^b$

Answer 1

You can use the fact that $x=a^{\log_a x}$, then $$ 3^7\times 4^8=(4^{\log_4 3})^7\times 4^8=4^{7\log_4 3}\times 4^8=4^{8+7\log_4 3}. $$

However, if you are asking whether there are non-trivial integer numbers $a$ and $b$, that $3^7\times 4^8=a^b$, then the answer is no.

Answer 2

You can always do this by a simple change of base. Suppose you have the product $a^mb^n,$ and you want to write as a single expression involving just one base, then change the base $b$ to a power of $a,$ so that we have that $b=a^{\log_ab}.$ Thus, we find that $$a^mb^n=a^m(a^{\log_ab})^n=a^ma^{n\log_ab}=a^{m+n\log_ab}.$$

In your example you would now have $$3^74^8=3^72^{16}=3^73^{16\log_32}=3^{7+16\log_32}.$$