Dumb Question: Why Do You Add Exponents When Multiplying?

[pandaexpress]

When multiplying x² and x⁵, we get x⁷. Quick, dumb question:

Why isn't it x¹⁰ instead (since we're multiplying and not adding)?

 

[jedishrfu]

Think of the exponent as counting the number of factors so x^2 is x*x and x^5 is x*x*x*x*x and when multiplied together gives you 7 x's or x^7

 

[Mark44]

When multiplying x² and x⁵, we get x⁷. Quick, dumb question:

Why isn't it x¹⁰ instead (since we're multiplying and not adding)?

##x^2## means ##x \cdot x##, and ##x^2## means ##x \cdot x \cdot x \cdot x \cdot x##, so ##x^2 \cdot x^5## means ##x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x##, right? How many factors of x are there in that last expression?

 

[pandaexpress]

Think of the exponent as counting the number of factors so x^2 is x*x and x^5 is x*x*x*x*x and when multiplied together gives you 7 x's or x^7

This might seem a little dumb, but I figured that multiplying would give you ¹⁰ still. Using your example above,

x² = x*x
x⁵ = x*x*x*x*x

Wouldn't x² * x⁵ = x*x; x*x; x*x; x*x; x*x? Basically, you'd have a pair of x*x's five times. Or, you would get:

x*x*x*x*x; x*x*x*x*x Or, in other words, two groups of x*x*x*x*x.

It's the same idea with regular multiplication, where if you have 2X5, then it's like

|| || || || || (five groups of two) or |||||; ||||| (two groups of five)

I just don't quite understand why we're adding and getting 7 (instead of 10), since the operation is to multiply exponents. Do you guys see what I mean?

If they want you to add the exponents, then why not just say x² + x⁵?

 

[Mark44]

This might seem a little dumb, but I figured that multiplying would give you ¹⁰ still. Using your example above,

x² = x*x
x⁵ = x*x*x*x*x

Wouldn't x² * x⁵ = x*x; x*x; x*x; x*x; x*x?

No.
What you have written is ##(x^2)^5##, or 5 factors of ##x^2##, making 10 factors of x, or ##x^{10}##.
With ##(x^2) \cdot (x^5)## you have two factors of x multiplying 5 factors of x, making 7 factors of x, or ##x^7##.

Basically, you'd have a pair of x*x's five times. Or, you would get:

x*x*x*x*x; x*x*x*x*x Or, in other words, two groups of x*x*x*x*x.

It's the same idea with regular multiplication, where if you have 2X5, then it's like

|| || || || || (five groups of two) or |||||; ||||| (two groups of five)

I just don't quite understand why we're adding and getting 7 (instead of 10), since the operation is to multiply exponents.

No, again. The operation is to multiply ##x^2## and ##x^5##, which results in the exponents being added.

Do you guys see what I mean?

If they want you to add the exponents, then why not just say x² + x⁵?

Because the terms in ##x^2 + x^5## are not like terms (i.e., same exponent), they can't be combined.

 

[DrewD]

Maybe try looking at it with numbers. For example, when ##x=2##, ##x^2\cdot x^5## is ##4\cdot32=128=2^7##. Try it with some more numbers. Now you will see that indeed, as Mark44 said, ##x^2\cdot x^5=(x\cdot x)\cdot(x\cdot x\cdot x\cdot x\cdot x)## and not what you wrote.