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Unit 4: Lesson 1

Meaning of exponents

Intro to exponents

How do we interpret exponents? 2 to the 3rd power (2^3) is the same as multiplying three 2's together: 2 x 2 x 2 = 8. So, in this case, the exponent (3) tells you how many times to multiply the base number (2) by itself. Created by Sal Khan.

Want to join the conversation?

• can you have an exponent that has a decimal or fraction like 5^4.8?
• Yes, it's like this:
4.8 = 4 4/5 = 24/5. So it's 5 √5 to the power of 24. So the answer is 1.
• what is 23 to the 42 power
• when your multiplying like 2 x 2 x 2 do u always have to put the dot for multiplication in that situation?
• In algebra, there are many different symbols, and if you write like 2 x 2 x 2, people will think it it is variable if you write it down like that. If you are not doing algebra, you could do that, but it is not ideal.
Thanks
• Do you mind specifying? I can't really understand your question.
• How do I divide exsponents I don't get it
• To divide exponents with the same base, keep the base and subtract the exponents:
3^5 / 3^3
5 - 3 = 2 (This is our new exponent)
• Can I do any number to the power of a negative number?
Like, 3^ -2 ??
• Yes, you can (though there is one limitation). It has to be a REAL Number!)
• When will you teach about square roots?
• Use the search bar at the top of any KA screen and search for "intro to square roots" to find the lesson.
• How is exponents like basically than Addition, Subtraction, Multiplication, and Division?
• Important to realize is that exponentiation is not simply repeated multiplication. Otherwise, what is 3^1.258? I can't multiply something 1.258 times, but this definitely has an answer! Similarly, if multiplication is repeated addition, then what is 6.2*5.5? I can't add a number half a time! If you think you can simply add half of 3.2 the last time, then let me ask you this. What would pi*pi be? Now, you have an infinite amount of decimals to worry about! You can never do this algebraically in terms of addition!

Multiplication is perhaps closer related to computing the area of a surface or (potentially hyperdimensional) volume based on its sides. Now pi * pi is the area of a surface formed by edges whose length is a rolled-out perimeter of a circle with unit diameter. There, we can understand pi * pi now!

Exponents are harder to understand in a simple, conceptual way. We can understand rational numbers quite easily using roots, but that leaves the question of irrational reals in between the rationals.

Although you could just write this off as voodoo-stuff that churns out a somewhat sensible
solution (perhaps using some higher-level math such as making sure the differentials all make sense). But
we usually define the value of raising bases to real-number or even complex-number exponents using the natural exponent using the fact that e^(ln(x)) = x which means x^z = (e^ln(x))^z = e^(ln(x)*z). Why this is convenient will become clear later when you work more with e and complex numbers!

Conclusion? Maths is fun, and it can be "easy" when you look for functional shortcuts such as "exponentiation is repeated multiplication, which is repeated addition". But the part where it gets "hard", although it requires a lot of effort to understand all of it, is much more rich, rewarding, and unveils a lot about how maths works at a higher level!