Tip:
Highlight text to annotate it
X
- WELCOME TO A LESSON ON THE 0 EXPONENT.
IF WE START BY LOOKING AT THE EXPONENT RULES
GIVEN HERE ON THE RIGHT, NOTICE THAT WE'RE GIVEN
THAT "A" TO THE 0 POWER = 1 AS LONG AS THE BASE "A"
DOESN'T = 0.
BEFORE WE TAKE A LOOK AT OUR EXAMPLES, THOUGH,
I THINK IT'S IMPORTANT THAT WE UNDERSTAND
WHY "A" TO THE 0 = 1.
SO, WE'RE GOING TO JUSTIFY THIS RULE SEVERAL WAYS
BEFORE WE TAKE A LOOK AT THESE EXAMPLES.
WE'LL FIRST TAKE A LOOK AT RAISING 2 TO VARIOUS POWERS
AND 3 TO VARIOUS POWERS.
WE'RE GOING TO START WITH 2 TO THE 4TH
AND THEN DECREASE THE EXPONENT BY 1 ALL THE WAY DOWN
TO 2 TO THE POWER OF -2.
WELL, WE SHOULD RECOGNIZE THAT 2 TO THE 4TH IS 2 x 2 x 2 x 2
OR 4 FACTORS OF 2, WHICH = 16.
AND, ALSO, 2 TO THE 3RD OR 2 x 2 x 2 = 8,
BUT NOTICE HOW THE PATTERN TO GO FROM 16 TO 8 WOULD BE
TO DIVIDE BY 2 SINCE 2 TO THE 3RD
HAS ONE LESS FACTOR OF 2.
2 TO THE 2ND OR 2 SQUARED = 2 x 2 OR 4.
NOTICE HOW 8 DIVIDED BY 2 ALSO = 4.
AND THEN WE HAVE 2 TO THE 1ST, WHICH = 2,
AND, AGAIN, NOTICE THAT 4 DIVIDED BY 2 = 2.
AND NOW WE HAVE 2 TO THE 0, WHICH BY DEFINITION = 1,
BUT ALSO NOTICE HOW THE PATTERN CONTINUES.
2 DIVIDED BY 2 = 1.
THIS ALSO EXPLAINS WHY NEGATIVE EXPONENTS
RESULT IN FRACTIONS.
2 TO THE POWER OF -1 = 1/2
BECAUSE 1 DIVIDED BY 2 = 1/2
AND 2 TO THE POWER OF -2 = 1/4.
NOTICE THAT 1/2 DIVIDED BY 2 = 1/4.
AND OF COURSE, THIS PATTERN WOULD BE TRUE
FOR ANY NON-ZERO BASE.
NOTICE, IF WE TAKE A LOOK AT THE BASE OF 3,
3 TO THE 4TH = FOUR FACTORS OF 3 OR 81,
AND 3 TO THE 3RD = 27.
NOTICE HOW 81 DIVIDED BY 3 = 27.
3 TO THE 2ND = 9, AND 27 DIVIDED BY 3 ALSO = 9.
3 TO THE 1ST = 3, AND 9 DIVIDED BY 3 IS ALSO 3.
AND HERE WE HAVE 3 TO THE 0 AGAIN,
WHICH WE KNOW = 1, BECAUSE 3 DIVIDED BY 3 = 1.
3 TO THE POWER OF -1 = 1/3, AND 1 DIVIDED BY 3 = 1/3.
AND THEN, FINALLY, 3 TO THE POWER OF -2 = 1/9,
AND 1/3 DIVIDED BY 3 = 1/9.
AND LET'S ALSO JUSTIFY THIS RULE
USING OUR PROPERTIES OF EXPONENTS.
FOR EXAMPLE, IF WE CONSIDER 2 RAISED TO THE POWER
OF 3 x 2 TO THE 0,
WE KNOW WHEN MULTIPLYING,
AND THE BASES ARE THE SAME, THE RULE IS TO ADD THE EXPONENTS.
SO IF WE ADD THE EXPONENTS, WE WOULD HAVE 2 RAISED
TO THE POWER OF 3 + 0, WHICH OF COURSE
JUST = 2 TO THE 3RD.
SO IF 2 TO THE 3RD x 2 TO THE 0 = 2 TO THE 3RD,
THE ONLY WAY THAT'S GOING TO BE TRUE IS IF 2 TO THE 0 = 1.
AND THEN IN THE SECOND ROW HERE WE HAVE THE SAME RULE
FOR ANY BASE "A" WHERE "A" DOESN'T EQUAL 0.
AND NOW FOR ONE MORE JUSTIFICATION,
LET'S CONSIDER 3 RAISED TO THE POWER OF 0,
WHICH MUST = 3 RAISED TO THE POWER OF -2 + 2,
BECAUSE -2 + 2 IS STILL 0, AND THEN BY APPLYING
THIS PRODUCT RULE HERE IN THE OPPOSITE DIRECTION,
WE CAN WRITE THIS AS A PRODUCT.
THIS WOULD BE 3 TO THE POWER OF -2 x 3 TO THE POWER OF +2,
AND THEN TO WRITE 3 TO THE POWER OF -2 WITH A POSITIVE EXPONENT,
WE WOULD MOVE IT TO THE DENOMINATOR.
SO THIS 3 TO THE POWER OF -2 = 3 TO THE 2ND IN THE DENOMINATOR,
WHICH, NOTICE, GIVES US 3 TO THE 2ND
OVER 3 TO THE 2ND OR 9/9, WHICH MUST = 1.
AND THEREFORE, 3 TO THE 0 MUST EQUAL 1.
AND THEN, AGAIN, THIS SECOND ROW HERE
IS JUST THE MOST GENERAL CASE, AND, AGAIN,
WHERE X WOULD NOT = 0.
SO, NOW, GOING BACK TO OUR EXAMPLES,
HOPEFULLY, YOU HAVE A BETTER UNDERSTANDING
WHY ANY NON-ZERO BASE RAISED TO THE 0 POWER MUST = 1.
AND AS WE GO THROUGH THESE, THERE IS ONE EXAMPLE WE HAVE
TO BE VERY CAREFUL ABOUT, AND THAT'S GOING TO BE
THIS LAST ONE.
AGAIN, 2 TO THE 0 = 1.
THEN, LOOKING AT THE SECOND EXAMPLE,
THERE'S NO REASON TO EVALUATE THIS INNER EXPRESSION
SINCE IT'S RAISED TO THE 0 POWER,
AND THEREFORE IT'S EQUAL TO 1.
4X TO THE 3RD RAISED TO THE 0 POWER MUST = 1,
AGAIN, AS LONG AS X DOESN'T = 0.
OUR FOURTH EXAMPLE, THE SAME THING.
THIS EXPRESSION IS RAISED TO THE 0 POWER
AS LONG AS THE EXPRESSION IS NOT 0, THIS = +1.
NEXT, WE HAVE -7 RAISED TO THE 0 POWER.
NOTICE HOW THE -7 IS IN PARENTHESES.
OUR BASE IS NON-ZERO, AND THEREFORE THIS = 1,
BUT FOR THIS LAST EXAMPLE, WE HAVE TO BE VERY CAREFUL,
BECAUSE THIS IS THE OPPOSITE OF +7 RAISED TO THE 0 POWER.
THIS EXPONENT OF 0 IS ONLY ATTACHED TO THE +7,
NOT THE -7, UNLIKE THE PREVIOUS EXAMPLE.
AND THEREFORE THIS IS GOING TO BE EQUAL TO THE OPPOSITE
OF 7 TO THE 0 POWER, WHICH IS ACTUALLY -1, NOT +1.
YOU CAN THINK OF THE NEGATIVE AS NOT BEING ATTACHED
TO THE EXPONENT, AND THEREFORE, THE RESULT IS GOING TO BE -1.
OKAY, I HOPE YOU FOUND THIS HELPFUL.