This document discusses operations with exponents such as addition, subtraction, and order of operations. It provides examples of adding terms with the same base but different exponents (a^m + a^n) and shows that they cannot be combined. It also gives examples of subtracting and multiplying terms with exponents through substitution to prove the rules. The document demonstrates applying order of operations, like first completing multiplication before addition when terms are combined.

1. Other Operations
of Numbers with
Exponents
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2. Adding Numbers with
Exponents
• You have to remember, that whenever you
are trying to add two numbers with
exponents together, you have to try it with
simple substitutions and then find the
relationships as they exist.

3. Substituting
a m + a m = 2a m
Now to proove this with substituting simple values
∴Let a = 2, m = 3, n = 1
Substitute the numbers into the formula
a m + a m = 2 3 + 2 3 = 8 + 8 = 16
2a m = 2 × 2 3 = 2 × 8 = 16
They are the same, so it proves the formula.

4. When the exponents are
different?
What about when the exponent
is not the same?
i.e. a m + a n Using the same values
i.e. a = 2, m = 3, n = 1
2 3 + 21 = 8 + 2 = 10
This ≠ 2 3
∴a m + a n = a m + a n
i.e. It can't be simplified further.

5. Subtracting Numbers in Index
Form
2ab 2 − ab 2 = ab 2
Why? Substitute the following:
a = 1, b = 2
2ab 2 = 8
ab = 4
2
8 − 4 = 4 which also equals ab 2

6. Applying Order of Operations
2x 2 + 3x 3 × x 2
First complete multiplication (or division)
3x3 × x 2 = 3x 5
∴ Answer = 2x 2 + 3x 5
The answer can’t be simplified further.

7. Applying Order of Operations
2x 2 + 3x 3 × x 2
First complete multiplication (or division)
3x3 × x 2 = 3x 5
∴ Answer = 2x 2 + 3x 5
The answer can’t be simplified further.