Simplify the Expression: a^4 × a^5 Using Exponent Rules

We will apply the law of exponents:

$a^m\cdot a^n=a^{^{m+n}}$

This means that when multiplying identical numbers raised to any power (meaning - identical bases raised to not necessarily identical powers), we can maintain the same base and simply add the exponents of the numbers,
Let's apply this law to the problem:

$a^4\cdot a^5=a^{4+5}=a^9$

Something important to remember is that this solution can also be explained verbally. Raising to a power effectively means multiplying the number (base) by itself as many times as the exponent indicates. Therefore multiplying $a$ by itself 4 times and multiplying the result by the result of multiplying $a$ by itself 5 times is like multiplying $a$ by itself 9 times, meaning multiplication between identical numbers (identical bases) raised to powers, not necessarily identical, can be calculated by keeping the same base (same number) and adding the exponents together.