2. Let’s review the basics of exponential notation.
Review on Exponents
3. base
exponent
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Let’s review the basics of exponential notation.
Review on Exponents
N times
4. base
exponent
Rules of Exponents
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Let’s review the basics of exponential notation.
Review on Exponents
N times
5. base
exponent
Multiply–Add Rule:
Rules of Exponents
Divide–Subtract Rule:
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Power–Multiply Rule:
Let’s review the basics of exponential notation.
Review on Exponents
N times
6. base
exponent
Multiply–Add Rule: ANAK = AN+K
Rules of Exponents
Divide–Subtract Rule:
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Power–Multiply Rule:
Let’s review the basics of exponential notation.
Review on Exponents
N times
7. base
exponent
Multiply–Add Rule: ANAK = AN+K
Rules of Exponents
Divide–Subtract Rule:
AN
AK = AN – K
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Power–Multiply Rule:
Let’s review the basics of exponential notation.
Review on Exponents
N times
8. base
exponent
Multiply–Add Rule: ANAK = AN+K
Rules of Exponents
Divide–Subtract Rule:
AN
AK = AN – K
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Power–Multiply Rule: (AN)K = ANK
Let’s review the basics of exponential notation.
Review on Exponents
N times
9. base
exponent
Multiply–Add Rule: ANAK = AN+K
Rules of Exponents
Divide–Subtract Rule:
AN
AK = AN – K
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Power–Multiply Rule: (AN)K = ANK
Let’s review the basics of exponential notation.
For example, x9x5 =x14 ,
x9
x5 = x9–5 = x4, and (x9)5 = x45.
Review on Exponents
N times
10. base
exponent
Multiply–Add Rule: ANAK = AN+K
Rules of Exponents
Divide–Subtract Rule:
AN
AK = AN – K
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Power–Multiply Rule: (AN)K = ANK
Let’s review the basics of exponential notation.
For example, x9x5 =x14 ,
x9
x5 = x9–5 = x4, and (x9)5 = x45.
Review on Exponents
N times
These particular operation–conversion rules appear often in
other forms in mathematics.
11. base
exponent
Multiply–Add Rule: ANAK = AN+K
Rules of Exponents
Divide–Subtract Rule:
AN
AK = AN – K
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Power–Multiply Rule: (AN)K = ANK
Let’s review the basics of exponential notation.
For example, x9x5 =x14 ,
x9
x5 = x9–5 = x4, and (x9)5 = x45.
Review on Exponents
N times
These particular operation–conversion rules appear often in
other forms in mathematics. Hence their names, the Multiply–
Add Rule, the Divide–Subtract Rule, the Power–Multiply Rule,
are important.
12. base
exponent
Multiply–Add Rule: ANAK = AN+K
Rules of Exponents
Divide–Subtract Rule:
AN
AK = AN – K
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN
Power–Multiply Rule: (AN)K = ANK
Let’s review the basics of exponential notation.
For example, x9x5 =x14 ,
x9
x5 = x9–5 = x4, and (x9)5 = x45.
Review on Exponents
N times
These particular operation–conversion rules appear often in
other forms in mathematics. Hence their names, the Multiply–
Add Rule, the Divide–Subtract Rule, the Power–Multiply Rule,
are important. Let’s extend the definition to negative and
fractional exponents.
14. Since = 1 = A1 – 1 = A0A1
A1
The Exponential Functions
15. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
The Exponential Functions
16. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
The Exponential Functions
17. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since =
1
AK
A0
AK
The Exponential Functions
18. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K,
1
AK
A0
AK
The Exponential Functions
19. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
The Exponential Functions
20. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Since (A )k = A = (A1/k )k,
k
The Exponential Functions
21. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k
The Exponential Functions
22. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Fractional Powers: A1/k = A.
k
The Exponential Functions
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k
23. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k
24. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Fractional Powers: A1/k = A.
k
The Exponential Functions
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
25. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Example A. Simplify.
b. 91/2 =
a.
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
9–2 =
c. 9 –3/2 =
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k
26. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Example A. Simplify.
1
92
1
81
b. 91/2 =
a.
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
9–2 = =
c. 9 –3/2 =
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k
27. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Example A. Simplify.
1
92
1
81
b. 91/2 = √9 = 3
a.
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
9–2 = =
c. 9 –3/2 =
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k
28. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Example A. Simplify.
1
92
1
81
b. 91/2 = √9 = 3
a.
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
9–2 = =
c. 9 –3/2 = (9½)–3
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k
Pull the numerator outside to
take the root and simplify the
base first.
29. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Example A. Simplify.
1
92
1
81
b. 91/2 = √9 = 3
a.
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
9–2 = =
c. 9 –3/2 = (9½)–3 = 3–3
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k
Pull the numerator outside to
take the root and simplify the
base first.
30. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the Negative Power Rule.
1
AK
A0
AK
Negative Power Rule: A–K =
1
AK
Example A. Simplify.
1
92
1
81
1
33
b. 91/2 = √9 = 3
a.
Fractional Powers: A1/k = A.
k
The Exponential Functions
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
9–2 = =
c. 9 –3/2 = (9½)–3 = 3–3 = = 1
27
Since (A )k = A = (A1/k )k, hence A1/k = A.
k k
Pull the numerator outside to
take the root and simplify the
base first.
32. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
33. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
34. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
Using fractional exponent notation, we write these steps as
if x3 = –8 then
35. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
Using fractional exponent notation, we write these steps as
if x3 = –8 then
x = (–8)1/3
The reciprocal of the power 3
36. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
Using fractional exponent notation, we write these steps as
if x3 = –8 then
x = (–8)1/3 = –2.
The reciprocal of the power 3
37. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
Using fractional exponent notation, we write these steps as
if x3 = –8 then
x = (–8)1/3 = –2.
(Rational) Power equations are equations of the type xP/Q = c.
The reciprocal of the power 3
38. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
Using fractional exponent notation, we write these steps as
if x3 = –8 then
x = (–8)1/3 = –2.
(Rational) Power equations are equations of the type xP/Q = c.
To solve them, we take the reciprocal power, that is,
if xP/Q = c,
The reciprocal of the power 3
39. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
Using fractional exponent notation, we write these steps as
if x3 = –8 then
x = (–8)1/3 = –2.
(Rational) Power equations are equations of the type xP/Q = c.
To solve them, we take the reciprocal power, that is,
if xP/Q = c,
then x = (±) c Q/P.
The reciprocal of the power 3
The reciprocal of the power P/Q
40. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
Using fractional exponent notation, we write these steps as
if x3 = –8 then
x = (–8)1/3 = –2.
(Rational) Power equations are equations of the type xP/Q = c.
To solve them, we take the reciprocal power, that is,
if xP/Q = c,
then x = (±) c Q/P.
Note that xP/Q may not exist, or that sometime we get both (±)
xP/Q solutions means that sometimes.
The reciprocal of the power P/Q
The reciprocal of the power 3
41. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
b. x2 = 64
c. x2 = –64
d. x –2/3 = 64
42. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3
b. x2 = 64
c. x2 = –64
d. x –2/3 = 64
43. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
x = √64 = 4.
b. x2 = 64
c. x2 = –64
d. x –2/3 = 64
44. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
c. x2 = –64
d. x –2/3 = 64
45. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
x = 641/2 or that
x = √64 = 8.
c. x2 = –64
d. x –2/3 = 64
46. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
x = 641/2 or that
We note that both ±8 are solutions.
x = √64 = 8.
c. x2 = –64
d. x –2/3 = 64
47. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
x = 641/2 or that
We note that both ±8 are solutions.
x = √64 = 8.
c. x2 = –64
x = (–64)1/2 which is UDF.
d. x –2/3 = 64
48. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
x = 641/2 or that
We note that both ±8 are solutions.
x = √64 = 8.
c. x2 = –64
x = (–64)1/2 which is UDF. (In fact what most calculators
return as the answer meaning that there is no real solutions.)
d. x –2/3 = 64
49. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
x = 641/2 or that
We note that both ±8 are solutions.
x = √64 = 8.
c. x2 = –64
x = (–64)1/2 which is UDF. (In fact what most calculators
return as the answer meaning that there is no real solutions.)
d. x –2/3 = 64
x = 64–3/2
50. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
x = 641/2 or that
We note that both ±8 are solutions.
x = √64 = 8.
c. x2 = –64
x = (–64)1/2 which is UDF. (In fact what most calculators
return as the answer meaning that there is no real solutions.)
d. x –2/3 = 64
x = 64–3/2
x = (√64)–3
51. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
x = 641/2 or that
We note that both ±8 are solutions.
x = √64 = 8.
c. x2 = –64
x = (–64)1/2 which is UDF. (In fact what most calculators
return as the answer meaning that there is no real solutions.)
d. x –2/3 = 64
x = 64–3/2
x = (√64)–3 = 8–3 = 1/512.
52. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first , then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
53. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first , then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
54. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first , then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2
x = (√4)3 = 8.
We note that both ±8 are solutions.
Mathematics Inputs in Text Format
Most digital calculation devices such as calculators, smart
phone apps or computer software accept inputs in the text
format. Besides the “+” , “–”, for addition and subtraction we
use “ * ” for multiplication, and “/” for the division operation.
The power operation is represented by “^”. For example, the
fraction is inputted as “3/4”, and the quantity 34 is “3^4”.
All executions of such inputs follow the order of operations.
3
4
55. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first , then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
56. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first , then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2
57. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first , then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2
x = (√4)3 = 8.
We note that both ±8 are solutions.
58. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first , then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2
x = (√4)3 = 8.
We note that both ±8 are solutions.
Mathematics Inputs in Text Format
Most digital calculation devices such as calculators, smart
phone apps or computer software accept inputs in the text
format.
59. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first , then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2
x = (√4)3 = 8.
We note that both ±8 are solutions.
Mathematics Inputs in Text Format
Most digital calculation devices such as calculators, smart
phone apps or computer software accept inputs in the text
format. Besides the “+” , “–”, for addition and subtraction we
use “ * ” for multiplication, and “/” for the division operation.
The power operation is represented by “^”.
60. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first , then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2
x = (√4)3 = 8.
We note that both ±8 are solutions.
Mathematics Inputs in Text Format
Most digital calculation devices such as calculators, smart
phone apps or computer software accept inputs in the text
format. Besides the “+” , “–”, for addition and subtraction we
use “ * ” for multiplication, and “/” for the division operation.
The power operation is represented by “^”. For example, the
fraction is inputted as “3/4”, and the quantity 34 is “3^4”.
All executions of such inputs follow the order of operations.
3
4
61. Power Equations and Calculator Inputs
Example B. Input and execute on a graphing
calculator or software.
Many common input mistakes happen for expressions involving
division or taking powers.
3
2
4
2 + 6
62. Power Equations and Calculator Inputs
Example B. Input and execute on a graphing
calculator or software.
Many common input mistakes happen for expressions involving
division or taking powers.
3
2
4
2 + 6
The correct text input is (2+6)/(4^(3/2)) to get the correct
answer of 1.
63. Power Equations and Calculator Inputs
Example B. Input and execute on a graphing
calculator or software.
Many common input mistakes happen for expressions involving
division or taking powers.
3
2
4
2 + 6
The correct text input is (2+6)/(4^(3/2)) to get the correct
answer of 1.
In general, when in doubt, insert ( )’s in the input to clarify the
order of operations.
