# Powers & Exponents (subtitle) [Pre-Algebra] I’ve got a postcard today 🙂 Guess what’s inside! This letter first started in the UK travels all around the world,
while giving huge fortunes to the recipients. This letter, now been transferred to you,
must leave your hand within a day. You must send seven copies of this letter to people who need luck. (…) Hmm, shall we start copying this letter? [Pre-Algebra]
[Powers and Exponents] If I send out 7 postcards today, 7 people will receive the postcards tomorrow. Now, supposing 7 people will dispatch 7 letters each, means 49 (=7×7) people will get the same mail
till the day after tomorrow. After three days, 7x7x7 people that are 343 will get the message. After four days, 7x7x7x7 people that are 2401 people will newly receive the same letter. 16807 people after five days, 117649 people after six days, and finally after nine days,
all people living in South Korea might get this divine letter. Anyway, multiplying the same numbers over and over again
was a quite confusing thing to do because it was hard to tell how many times the number is being multiplied. Is there any better way of expressing these multiplications neatly
not by enumerating all the numbers? Now, let’s talk about the way of writing repeated multiplication in a simpler way. Called a “power.” The power is a simple expression to represent
repeated multiplication of the same number or factor. A power is consists of the “base,”
which indicates the number that is multiplied and the “exponent,” which tells
how many times the base number is used as a factor. For example,or has the “base” number 7 at the bottom and the “exponent” 2 as a small number
written above and to the right of the base number. Now, let’s handle the fortune letter problem
in a shorter way using the powers. (7×7) multiplies two sevens, which means 7 is the base, and 2 is the exponent.
That we callor. Similarly, (7x7x7) multiplies three sevens, which can be shortened by the base 7, and the exponent 3.
We read the number 7^3 asor. Again, (7x7x7x7) has four sevens and we simplify the multiplication
asor. Likewise, we say “7 to the fifth power” or “7 to the fifth,”
“7 to the sixth power” or “7 to the sixth,” “7 to the seventh power” or “7 to the seventh,”
“7 to the eighth power” or “7 to the eighth,” and “7 to the n th power” or “7 to the n th.” It’s time to solve some examples! — Q. Choose the wrong equations. (2 answers) — 1) 3 x 3 x 5 x 7 x 7 x 7=3^2 x 5 x 7^3 ? How many threes do we have?
Right, two. Also, we have one five, and three sevens. Therefore, we can shorten the multiplication as
(3 squared) x 5 x (7 cubed) That is a correct equation. — 2) Is 10 squared equals to 20? In the 10 squared,
how many times does the multiplication of 10 repeats? Twice. Right? Therefore, 10 squared is nothing but 10 times 10,
which is 100. Wrong equation! Let’s move on. — 3) Is (1/5 x 1/5 x 1/5) equals to 3 divided by 5^3? On the left-hand side, 1 divided by 5 is multiplied three times,
which is 1/5 cubed. Or, you can say three fives are multiplied in the denominator
and three ones are multiplied in the numerator. That is 1 over 5 to the third power. By the way, in the numerator, multiplying three 1s gives 1. No matter how many times you multiply ones together,
the result is still one, you know. Therefore, equation (3) is also incorrect! Let’s take a look at the equations (4) and (5) too. — 4) The four squared in the numerator can be expanded to 4 x 4, where 4 can be expressed as 2 x 2. Simplifying 2 x 2 x 2 x 2 x 2 using a power gives “2 to the fifth”. Therefore, equation number (4) is correct. In equation (5), we have two 11s and three 13s in the numerator. That is 11 squared times 13 cubed. Equation (5) is also correct. To sum up, A power is a simplified expression
for multiplying the same numbers or factors over and over again. A base number tells what number or factor is being multiplied. An exponent, a superscript,
tells how many times the base is being multiplied. Now, when you finish this lecture,
you should text me like “ha^5”, rather than “hahahahaha!” Before finish up there are two more things to mention. A. One raised to any power is one.
(No matter how many times you multiply 1s, the result is still 1) B. When an exponent is one, it is simply the base itself. I mean, any number raised to the first power is that number. Thank you 🙂 Thank you 😀