Final answer:
In adding and subtracting scientific notation, the same power of 10 is needed to make the exponents the same. In multiplying and dividing scientific notation, different powers of 10 are allowed because the resulting number has a power of 10 that reflects the combined effect of the multiplication or division.
Step-by-step explanation:
In adding and subtracting scientific notation, you need the same power of 10 because the goal is to have the numbers represented in a form where all the exponents have the same value. This allows for easier addition or subtraction of the numbers. For example, to add 3.2 x 104 and 1.8 x 104, you need to rewrite them in the same power of 10:
3.2 x 104 + 1.8 x 104 = 3.2 x 104 + 0.18 x 105
Then, you can add the numbers out front and keep the same power of 10:
3.2 x 104 + 0.18 x 105 = 3.38 x 104
In multiplying and dividing scientific notation, you can have different powers of 10. This is because when multiplying, you multiply the numbers out front and then add the exponents. When dividing, you divide the numbers out front and then subtract the exponents. The resulting number will have a power of 10 that reflects the combined effect of multiplying or dividing the original powers of 10.
For example, to multiply 3.2 x 104 by 1.8 x 105, you multiply the numbers out front (3.2 x 1.8) and add the exponents (4 + 5):
(3.2 x 104) x (1.8 x 105) = (3.2 x 1.8) x 104+5 = 5.76 x 109