Exponent Algebra: Why is it that: x^5 + x^2 = x^5 + x^2, not x^7; 3x^p + 5x^{2p} = 3x^p + 5x^{2p}, not 8x^{2p}
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What I think's happening here is that you're simply trying to apply a list of rules you learned by rote, without understanding why the rules are what they are. It may help you think about what integer exponents really mean in terms of repeated multiplication. We can rewrite your first example as:
\(\displaystyle (x \cdot x \cdot x \cdot x \cdot x) + (x \cdot x)\)
Compare this with what you claim it's equal to (\(x^7\)):
\(\displaystyle (x \cdot x \cdot x \cdot x \cdot x) \cdot (x \cdot x)\)
Is there any reason to believe that these two things would be the same? In fact, we can use this example to see that the rule is actually that we can add exponents straight across when powers are being multiplied, which makes perfect sense when we think about what exponents mean: (5 copies of x) times (2 copies of x) definitely equals (7 copies of x).
\(\displaystyle (x \cdot x \cdot x \cdot x \cdot x) + (x \cdot x)\)
Compare this with what you claim it's equal to (\(x^7\)):
\(\displaystyle (x \cdot x \cdot x \cdot x \cdot x) \cdot (x \cdot x)\)
Is there any reason to believe that these two things would be the same? In fact, we can use this example to see that the rule is actually that we can add exponents straight across when powers are being multiplied, which makes perfect sense when we think about what exponents mean: (5 copies of x) times (2 copies of x) definitely equals (7 copies of x).