Product rule of exponents

The product rule of exponents states that $x^a \times x^b = x^{a + b}$.
It works because $x^a$ means $a$ copies of $x$ multiplied together and $x^b$ means $b$ more, giving $a + b$ copies in total.

So when the base is the same, you just add the exponents.
For $5^2 \times 5^3$, the base is $5$, so the result is $5^{2 + 3} = 5^5$.
There is no need to work out the actual numbers first.

The rule applies only when the bases match.
In $3^4 \times 5$, the bases $3$ and $5$ are different, so the expression cannot be simplified and stays as $3^4 \times 5$.
You can only add exponents that sit on the same base.

The same idea works with variables and with negative exponents.
To isolate $y$ in $\frac{y}{x^2} = x^3$, multiply both sides by $x^2$, giving $y = x^3 \times x^2 = x^{5}$.
And a symbolic power like $a^{n + 2}$ can be split apart as $a^n \times a^2$ whenever that is convenient.

Simplify $2^3 \times 2^4$.
The base is $2$ on both, so add the exponents: $2^{3 + 4}$.
That is $2^7$.

Solve for $y$: $\frac{y}{x^2} = x^3$.
Multiply both sides by $x^2$: $y = x^3 \times x^2$.
Adding the exponents, $y = x^{5}$.

Can $7^2 \times 4$ be simplified using the product rule?
The bases $7$ and $4$ are different, so their exponents cannot be added.
So it stays as $7^2 \times 4$, which is $49 \times 4 = 196$.