Final Answer:
To multiply powers with the same base, add the exponents.
To divide powers with the same base, subtract the exponent of the divisor from the exponent of the dividend.
Step-by-step explanation:
When multiplying powers with the same base, we are essentially multiplying the number of times that base is raised to a certain power. For example, if we have 2 raised to the power of 3 (2³) and 2 raised to the power of 4 (2⁴), we can see that both expressions involve raising the number 2 to a certain power.
To find out how many times we are raising 2 in total when we multiply these expressions, we add the exponents: 3 + 4 = 7. Therefore, when we multiply these expressions, we get 2 raised to the power of 7 (2⁷).
When dividing powers with the same base, we are essentially finding out how many times a certain power is contained within another power. For example, if we have 8 raised to the power of 3 (8³) and we want to divide this by 8 raised to the power of 2 (8²), we can see that both expressions involve raising the number 8 to a certain power.
To find out how many times we are raising 8 in the divisor expression when compared to the dividend expression, we subtract the exponent of the divisor from the exponent of the dividend: 3 - 2 = 1. Therefore, when we divide these expressions, we get 8 raised to the power of 1 (8¹).
In summary, when multiplying or dividing powers with the same base, we add or subtract exponents respectively to simplify our expressions and make calculations easier.