INDICES AND POWERS Flashcards
POSITIVE POWERS
HOW IS A NUMBER TO A POWER WRITTEN?
A NUMBER WRITTEN AS A SUPERSCRIPT AFTER A NUMBER.
POSITIVE POWERS
WHAT ARE THE OTHER NAMES FOR A POWER?
- POWER
- INDICES
- EXPONENT
POSITIVE POWERS
WHAT IS THE NUMBER TO THE LEFT OF THE POWER CALLED?
BASE
POSITIVE POWERS
HOW DOES TWO TO THE POWER OF TWO LOOK?
2²
POSITIVE POWERS
WHAT DOES 2² MEAN?
TWO SQUARED
2 X 2
POSITIVE POWERS
HOW DOES TWO TO THE POWER OF THREE LOOK?
2³
POSITIVE POWERS
WHAT DOES 2³ MEAN?
TWO CUBED
2 X 2 X 2
POSITIVE POWERS
CAN THE BASE AND THE POWER BE WRITTEN AS VARIABLES?
YES
POSITIVE POWERS
CAN BOTH THE BASE AND THE POWER NUMBER BE VARIABLES?
YES
(y^x)
POSITIVE POWERS
FIND THE VOLUME OF A SPHERE OF 6CM
- RADIUS r = DIAMETER / 2
- 6/2 = 3
- VOLUME v = 4/3 X πr³
- = 4/3 X 3.142 X 3³
- = 4 X 3.142 X 3 X 3
- = 113.04 cm³
NEGATIVE POWERS
HOW CAN WE WORK OUT A BASE RAISED TO A NEGATIVE POWER?
- THE BASE MUST BE RAISED TO THE SAME POSITIVE POWER
- THE RESULT MUST THEN BE INVERTED
- THE ONLY EXCEPTION IS THAT THE BASE MUST NOT BE ZERO
NEGATIVE POWERS
HOW CAN WE INVERT A NUMBER?
DIVIDE ONE BY THAT NUMBER
NEGATIVE POWERS
HOW WOULD TWO TO THE POWER OF MINUS TWO LOOK?
2^-2
NEGATIVE POWERS
WHAT IS 2^-2 EQUAL TO?
1/(2X2) = 1/4
NEGATIVE POWERS
WHAT IS 2^-3 EQUAL TO?
1/ 2 X 2 X 2 = 1/8
NEGATIVE POWERS
WHAT IS THE GENERAL FORMULA FOR NEGATIVE POWERS?
a^-m = 1/a^m
a ≠ 0
NEGATIVE POWERS
FIND THE VALUE OF (3/2)^-3
- (3/2) ^-3 = (2/3)³ = 2³/3³ = 8/27
- OR
1. (3/2)³ = 3³/ 2³ = 27/8
2. (3/2)^-3 = 1/(3/2)³ = 1/ 27/8 = 8/27
NEGATIVE POWERS
- THE PRESSURE P EXERTED ON THE GROUND BY A SQUARE PLATE OF SIDE x SUPPORTING A WEIGHT w IS GIVEN BY
- P = wx^-2
- FIND THE VOLUME OF P IN PASCALS WHEN
- w = 300N AND x = 10cm
- x = 10cm = 0.1m
- P = wx^-2
- = 300 x 0.1^-2
- = 300 / 0.1²
- P = 30,000 Pa
FRACTIONAL POWERS
WHAT ELSE CAN A POWER BE IF NOT A NUMBER?
- A FRACTION
- A DECIMAL
FRACTIONAL POWERS
WHEN EXPRESSED AS A FRACTION WHAT DOES THE DENOMINATOR INDICATE?
THE ROOT OF THE NUMBER
FRACTIONAL POWERS
WHAT DOES 4^1/2 MEAN?
- BASE NUMBER - 4
- POWER - 1/2
- 4 TO THE POWER OF A HALF IS EQUAL TO THE SQUARE ROOT OF 4
- 4^1/2 = 2√4 = 2
FRACTIONAL POWERS
WHAT DOES 8^1/3 MEAN?
- 8 TO THE POWER OF A THIRD
- IT IS EQUAL TO THE CUBE ROOT OF 8
- 8^1/3 = 3√8 = 2
FRACTIONAL POWERS
WHAT DOES 8^0.33 MEAN?
- 8 TO THE POWER OF A THIRD
- IT IS EQUAL TO THE CUBE ROOT OF 8
FRACTIONAL POWERS
HOW ARE POWERS WRITTEN IF MADE UP OF AN INTEGER AND A FRACTIONAL PART?
- (8²)^1/3
- OR
- 3√8²
FRACTIONAL POWERS
CAN POWERS THAT ARE MADE UP OF AN INTEGER AND A FRACTIONAL PART BE WRITTEN AS A MIXED NUMBER?
NO, NEVER
FRACTIONAL POWERS
HOW DO YOU WORK OUT AN IMPROPER FRACTION?
- EACH PART OF THE FRACTION MAY BE APPLIED TO THE BASE IN SEQUENCE
- THE NUMBERATOR AS A POSITIVE OR NEGATIVE POWER
- THE DENOMINATOR AS A FRACTIONAL POWER
FRACTIONAL POWERS
WHAT DOES 8^2/3 MEAN?
- 8 TO THE POWER OF TWO THIRDS
- IT IS EQUAL TO
- (8²)^1/3
- OR
- ³√8²
FRACTIONAL POWERS
HOW WOULD YOU WORK OUT
³√8²?
- THE BASE NUMBER IS SQUARED
- 8² = 64
- THE CUBE ROOT IS THEN TAKEN
- ³√64 = 4
FRACTIONAL POWERS
HOW WOULD YOU WORK OUT 8^-2/3?
- INVERSE THE OPERATION
- 8^-2/3
- 1/8^2/3 = 1/4
FRACTIONAL POWERS
- A NON-DIMENSIONAL CONSTANT k USED IN CONNECTION WITH RECTANGULAR WEIRS IS GIVEN BY
- k = H^2/3g^1/2 / n
- FIND THE VALUE OF k WHEN
- H = 4.56g
- g = 9.81
- n = 8.42 x 10^-6
- SEE PICTURE ATTACHED
SPECIAL CASES
WHAT IS THE RULE FOR A BASE RAISED TO THE POWER OF ZERO?
- ANY BASE RAISED TO THE POWER OF ZERO IS EQUAL TO 1
- THE BASE CAN NOT BE 0
- a^0 = 1
- a ≠ 0
SPECIAL CASES
WHAT IS THE RULE FOR A BASE RAISED TO THE POWER OF 1?
- ANY BASE RAISED TO THE POWER OF 1 IS EQUAL TO THE VALUE OF THE BASE
- a¹ = a
POWERS IN MUTIPLICATION AND DIVISION
WHAT CAN YOU DO IF MULTIPLYING TWO LIKE BASE NUMBERS WITH POWERS?
- THE POWERS CAN BE ADDED
- ONLY IF THE BASE NUMBERS ARE THE SAME
- 2³ x 2² = 2^5
- (2 x 2 x 2) x (2 x 2)
- 2 x 2 x 2 x 2 x 2
- 2^5
POWERS IN MULTIPLICATION AND DIVISION
WHAT CAN YOU DO IF DIVIDING TWO LIKE BASE NUMBERS WITH POWERS?
- IF THE BASES ARE THE SAME THE POWERS MAY BE SUBTRACTED
- 5^7 / 5^3 = 5 x 5 x 5 x 5 x 5 x 5 x 5 / 5 x 5 x 5
- 7 - 3 = 4
- 5 x 5 x 5 x 5
- = 5^4
LAWS OF INDICES
MULTIPLICATION
ADD EXPONENTS
LAWS OF INDICES
DIVISION
SUBTRACT EXPONENTS
LAWS OF INDICES
POWER OF A POWER
MULTIPLY EXPONENTS
LAWS OF INDICES
POWER OF 1
THE TERM ITSELF
LAWS OF INDICES
POWER OF 0
EQUALS 1
LAWS OF INDICES
NEGATIVE INDICES
TAKE THE RECIPROCAL
LAWS OF INDICES
FRACTIONAL INDICES
NUMERATOR BECOMES THE EXPONENT
DENOMINATOR BECOMES THE ROOT
