Question

Why, in adding and subtracting scientific notation, do you need the same power of 10 whereas in multiplying and dividing scientific notation, you can have two different powers of 10?

A) Addition and subtraction involve combining terms, requiring the same units.
B) Multiplication and division involve scaling quantities, allowing for different units.
C) Addition and subtraction involve multiplication, while multiplication and division involve addition.
D) Multiplication and division do not require consistent units, but addition and subtraction do.

Answer 1

In adding and subtracting scientific notation, the same power of 10 is required because we are combining like terms, which need the same units or scale. In contrast, during multiplication and division, different powers of 10 can be managed through the properties of exponents, where coefficients are multiplied or divided and exponents are added or subtracted.

Step-by-step explanation:

The reason we need the same power of 10 for adding and subtracting scientific notation is linked to how we combine like terms in basic arithmetic. These operations require the terms to have the same units or scale, just as you would need to have apples compared with apples and not oranges. Conversely, when we are multiplying and dividing in scientific notation, we are scaling numbers, and different powers can be easily managed through the properties of exponents. With multiplication, we multiply the coefficients (the numbers in front of the power of ten) and add the exponents. In division, we divide the coefficients and subtract the exponents.

For example, when adding 2 x 103 + 3 x 103, the exponents need to be the same because we are combining the values of N (which in this case are 2 and 3). However, for (3 x 105) x (2 x 100), we simply multiply 3 by 2 to get 6, and add the exponents 5 and 0 to get 6 x 105.