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Transcript of Exponents
La rouge had started his exponent theory in 1863. His logic was to multiply the numbers over and over again so that he made sure his answers were accurate.
He created the exponent theory to solve the problem of multiplying a number repeatedly.
The power is normally confused as the exponent because people think that power means the amount of times you multiply it. However, the exponent is the number that the base is raised to, and the power is the entire number.
Introduction to Exponents
What are exponents and
how are they written?
The exponent of a number says how many times the base is being multiplied by itself. For example, in 4^2, the two says you multiply the four twice.
An exponent tells you how many times the base number is used as a factor. A base of five raised to the second power is called "five squared" and basically means "five times five".
Real Life Question
You are offered a job at Burger King that pays 1 cent on the first day, 2 cents on the second day, 4 cents on the third day and 8 cents on the fourth day and so on for 30 days. How much will you be paid in total if you worked 30 days?
We have to use exponents to solve this. We could write them all out and solve it, but that would defeat the purpose of exponents.
Why do we need exponents?
Exponents are a shorthand notation for multiplying the same number by itself several times.
It is easier to write 3^16 rather than writing 3*3*3*3*3*3*3*3*3*3*3*3*3*3*3*3.
Exponents are a simple way to write and multiply the same number multiple times.
They’re simpler than doing mathematical equations long-handed.
By: Seana, Araiya, and Chantelle
The Product Rule states that when multiplying powers with the same base, you can add the exponents to get an accurate and shorter answer.
This is a commonly confused rule because some people think you add the powers with the same base and multiply the exponent, which is incorrect. Instead of multiplying the exponents, you add them. When you have a different bases, you multiply the base and add the exponents .
Example 1: 9^10 + 9^15 = 9^25
Example 2: 10^4 + 10^6 = 10^10
Example: (different bases): 5^8 + 3^1= 15^9
Incorrect Example : 12^5 + 12^3 = 12^15. This is incorrect because when you have the same base, you have to add the exponents, not multiply them.
The Power Rule states that to raise a power to a power, you multiply the exponents.
It looks like this: (x^4)3 = x^4 • 3 = x^12.
For example, 5^2 raised to the third power is equal to 5^6. Similarily, 3^2 raised to the third power is 3^6.
Some people mix up the Power and the Product Rules.
The Power Rule is when a power is raised to another power and the two powers are multiplied. For example: (2^3)2 = 2^3·2 = 2^6.
The Product Rule does not have two raised exponents, only one, then you add the two exponents. For example: 3^3 + 3^6 = 3^9.
CORRECT AND INCORRECT EXAMPLES
Example 1: (6^4)4 = 6^4 • 4 = 6^16
Example 2: (4^3)4 = 4^3 • 4 = 4^12
Incorrect Example: (23)4 = 23 + 4 = 27.
This is an incorrect example because in the power rule, you are multiplying-not adding-the raised powers. Adding the powers is known as the Product Rule.
Negative Power Rule
The Zero Exponent Rule states that any nonzero number raised to a number zero will be one. This is so because the number is multiplied by itself zero times.
However, if the zero is raised to another zero, the answer is zero (nothing).
A negative exponent of a number equals to the reciprocal of a positive power of a number.
The quotient rule states that when dividing powers with the same base, you can keep the base and subtract the exponents.
For example, in the problem 8^4/8^2, you could subtract the 2 from the 4 (the exponents) to make 8^2, instead of multiplying both the powers out and dividing the answers. If you did that, you would have to do: 8*8*8*8/8*8.
In the diagram below, the variable
is being raised to a negative exponent. If it's negative, it changes to a fraction,
now being multiplied by the same exponent. After you multiply the exponent, you can either leave the fraction like that or divide it: 1/x^n.
10^-3 = 1/10^3 = 1/1000 = 0.001
The Sum of Consecutive Powers
We'll use the sum of consecutive powers to solve it.
Look for the pattern in the numbers.
The salary is doubling by two every day that you work.
So the exponent has to be 2.
Let x = total amount of money for 30 days
x = 1 + 2^1 + 2^2 + 2^3 + 2^4... 2^29
It is 2^29 because on day one, you're being paid 1 cent, which is the one in the equation. If you count it, 1 + 2^1, 2^2, 2^3... all the way equals 30 days.
So, now that we've lost the 1 after we multiplied it by 2, we'll add it back to the equation.
2x + 1 = (1 + 2^1 + 2^2 + 2^3 + 2^4.... 2^29) + 2^30
Notice the part in parenthesis (only in parenthesis to show where it is, that's not how you solve the problem) is the same thing that x is equal to above.
So, we can replace that big section with x.
So, now the equation is much shorter.
2x + 1 = x + 2^30
Now we combine the like terms by bringing the variables to one side (left) and the numbers to the other (right).
2x - x = 2^30 - 1
2x - x is the same thing as 2 - 1, so 2x - x equals x.
x = 2^30 - 1
2^30 = 1073741824 cents
To change it to dollars, move the decimal point to places to the left (divide by 100).
x = $10,737, 418.24 - 1
x = $10,737, 417.24
Sometimes you have to make the problem bigger to make it smaller (easier). So we'll multiply the equation on both sides by 2.
2x = 2^1(2^0 + 2^1 + 2^3 + 2^4 + 2^5... 2^30)
2 = 2^1, and when multiplying powers with the same base, you add the exponents (Product Rule). 2^0 = 1.