Power Operations & Scientific Notation, Lecture notes of Algebra

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Power Operations &

Scientific Notation

1. Power Operations

Powers are also called exponents or indices ; we can work with the indices to simplify expressions and to solve problems.

Some key ideas:

a) Any base number raised to the power of 1 is the base itself: for example, 51 = 5 b) Any base number raised to the power of 0 equals 1, so: 40 = 1 c) Powers can be simplified if they are multiplied or divided and have the same base. d) Powers of powers are multiplied. Hence, (2^3 )^2 = 2^3 × 2^3 = 2^6 e) A negative power indicates a reciprocal: 3 −2^ = (^312)

Certain rules apply and are often referred to as: Index Laws.

Below is a summary of the index rules:

Index Law Substitute variables for values 𝒂𝒂 𝒎𝒎^ × 𝒂𝒂 𝒏𝒏^ = 𝒂𝒂 𝒎𝒎+𝒏𝒏^23 × 2^2 = 23+2^ = 2^5 = 32 𝒂𝒂 𝒎𝒎^ ÷ 𝒂𝒂 𝒏𝒏^ = 𝒂𝒂 𝒎𝒎−𝒏𝒏^36 ÷ 3^3 = 36−3^ = 3^3 = 27 (𝒂𝒂 𝒎𝒎)𝒏𝒏^ = 𝒂𝒂 𝒎𝒎𝒏𝒏^ (4^2 )^5 = 4^2 ×^5 = 4^10 = 1048 576 (𝒂𝒂𝒂𝒂)𝒎𝒎^ = 𝒂𝒂 𝒎𝒎𝒂𝒂𝒎𝒎^ (2 × 5)^2 = 2^2 × 5^2 = 4 × 25 = 100 (𝒂𝒂^ �𝒂𝒂 )𝒎𝒎^ = 𝒂𝒂 𝒎𝒎^ ÷ 𝒂𝒂𝒎𝒎^ (10 ÷ 5)^3 = 2^3 = 8; (10^3 ÷ 5^3 ) = 1000 ÷ 125 = 8

𝒂𝒂 −𝒎𝒎^ =

𝒂𝒂 𝒎𝒎^

4 −2^ =

𝟏𝟏 𝒎𝒎 (^) = 𝒎𝒎√𝒂𝒂^8 (^1) � (^3) = √ 8 3 = 2 𝒂𝒂 𝟎𝟎^ = 𝟏𝟏 63 ÷ 6^3 = 63−3^ = 6^0 = 1; (6 ÷ 6 = 1)

E XAMPLE P ROBLEMS :

a) Simplify 65 × 6^3 ÷ 6^2 × 7^2 + 6^4 = = 65+3−2^ × 7^2 + 6^4

= 6^6 × 7^2 + 6^4

b) Simplify 𝑔𝑔 5 × ℎ 4 × 𝑔𝑔 −1^ = = 𝑔𝑔 5 × 𝑔𝑔 −1^ × ℎ 4

= 𝑔𝑔 4 × ℎ 4

Watch this short Khan Academy video for further explanation: “Simplifying expressions with exponents” https://www.khanacademy.org/math/algebra/exponent-equations/exponent-properties- algebra/v/simplifying-expressions-with-exponents

2. Scientific Notation

Numbers as multiples or

fractions of ten

Number Number as a power of ten

10 x 10 x 10 1000 103

10 x 10 100 102

10 x 1/10 1 100

1/10 0.1 10 −^1

1/100 0.01 10 −^2

1/1000 0.001 10 −^3

Scientific notation is a convenient method of representing and working with very large and very small numbers. Transcribing a number such as 0.000000000000082 or 5480000000000 can be frustrating since

there will be a constant need to count the number of zeroes each time the number is used. Scientific notation provides a way of writing such numbers easily and accurately. Scientific notation requires that a number is presented as a non-zero digit followed by a decimal point and

then a power (exponential) of base 10. The exponential is determined by counting the number places the decimal point is moved.

The number 65400000000 in scientific notation becomes 6.54 x 10^10. The number 0.00000086 in scientific notation becomes 8.6 x 10-^.

(Note: 10-6^ = 1016 .)

If n is positive, shift the decimal point that many places to the right. If n is negative, shift the decimal point that many places to the left.

Question 2:

Write the following in scientific notation: a. 450

b. 90000000

c. 3.

d. 0.

Write the following numbers out in full: e. 3.75 × 10^2

f. 3.97 × 10^1

g. 1.875 × 10−

h. −8.75 × 10−

(Coefficient)

(base 10)

3. Calculations with Scientific Notation

Multiplication and division calculations of quantities expressed in scientific notation follow the

index laws since they all they all have the common base, i.e. base 10.

Here are the steps:

Multiplication Division A. Multiply the coefficients 1. Divide the coefficients B. Add their exponents 2. Subtract their exponents C. Convert the answer to scientific Notation

3. Convert the answer to scientific Notation Example: �𝟕𝟕. 𝟏𝟏 × 𝟏𝟏𝟎𝟎−𝟒𝟒^ � (^) × (𝟖𝟖. 𝟓𝟓 × 𝟏𝟏𝟎𝟎−𝟓𝟓^ ) 𝟕𝟕. 𝟏𝟏 × 𝟖𝟖. 𝟓𝟓 = 𝟔𝟔𝟎𝟎. 𝟑𝟑𝟓𝟓 (multiply coefficients) 𝟏𝟏𝟎𝟎−𝟒𝟒^ × 𝟏𝟏𝟎𝟎−𝟓𝟓^ = 𝟏𝟏𝟎𝟎(−𝟒𝟒+^ −𝟓𝟓)=−𝟗𝟗^ (add exponents) = 𝟔𝟔𝟎𝟎. 𝟑𝟑𝟓𝟓 × 𝟏𝟏𝟎𝟎−𝟗𝟗^ – check it’s in scientific notation  = 𝟔𝟔. 𝟎𝟎𝟑𝟑𝟓𝟓 × 𝟏𝟏𝟎𝟎−𝟖𝟖^ – convert to scientific notation 

Example: (9 × 10^20 ) ÷ (3 × 10^11 ) 9 ÷ 3 = 3 (divide coefficients) 1020 ÷ 10^11 = 10 (20−11)=9^ (subtract exponents) = 3 × 10^9 – check it’s in scientific notation 

Recall that addition and subtraction of numbers with exponents (or indices) requires that the base and the exponent are the same. Since all numbers in scientific notation have the same base 10, for

addition and subtraction calculations, we have to adjust the terms so the exponents are the

same for both. This will ensure that the digits in the coefficients have the correct place value so they can be simply added or subtracted.

Here are the steps:

Addition Subtraction

1. Determine how much the smaller exponent must be increased by so it is equal to the larger exponent 1. Determine how much the smaller exponent must be increased by so it is equal to the larger exponent
2. Increase the smaller exponent by this number and move the decimal point of the coefficient to the left the same number of places 2. Increase the smaller exponent by this number and move the decimal point of the coefficient to the left the same number of places
3. Add the new coefficients 3. Subtract the new coefficients
4. Convert the answer to scientific notation 4. Convert the answer to scientific notation Example: �𝟑𝟑 × 𝟏𝟏𝟎𝟎𝟐𝟐^ � (^) + (𝟐𝟐 × 𝟏𝟏𝟎𝟎𝟒𝟒^ ) 𝟒𝟒 − 𝟐𝟐 = 𝟐𝟐 increase the small exponent by 2 to equal the larger exponent 4 𝟎𝟎. 𝟎𝟎𝟑𝟑 × 𝟏𝟏𝟎𝟎𝟒𝟒^ the coefficient of the first term is adjusted so its exponent matches that of the second term = �𝟎𝟎. 𝟎𝟎𝟑𝟑 × 𝟏𝟏𝟎𝟎𝟒𝟒^ � + �𝟐𝟐 × 𝟏𝟏𝟎𝟎 𝟒𝟒^ � the two terms now have the same base and exponent and the coefficients can be added = 𝟐𝟐.𝟎𝟎𝟑𝟑 × 𝟏𝟏𝟎𝟎𝟒𝟒^ check it’s in scientific notation 

Example: (5.3 × 10^12 ) (^) − (4.224 × 10^15 ) 15 − 12 = 3 increase the small exponent by 3 to equal the larger exponent 15

1. 0053 × 10^15 the coefficient of the firs t term is adjusted so its exponent matches that of the second term = (0.0053 × 10^15 )^ − (4.224 × 10^15 ) the two terms now have the same base and exponent and the coefficients can be subtracted. = − 4. 2187 × 1015 check it’s in scientific notation 

4 Answers

Q1. Power Operations

a) i. 52 × 5^4 + 5^2 = 5^6 + 5^2 ii. x^2 × x^5 = x^7 iii. 42 × t^3 ÷ 4^2 = t^3 iv. (5^4 )^3 = 512

v. 24 36 34 =^2

432 = 16 × 9 = 144

vi. 32 × 3−5^ = 3−3^ = 1 27

vii. 9 (𝑥𝑥 2 )^3 3𝑥𝑥𝑥𝑥 2 =^

9𝑥𝑥 6 3𝑥𝑥𝑥𝑥^2 =^

3𝑥𝑥 5 𝑥𝑥^2

viii. 𝑎𝑎 −1^ √𝑎𝑎 = 𝑎𝑎 −1^ × 𝑎𝑎

1 (^2) = 𝑎𝑎 −

1 (^2) = 1 √𝑎𝑎^ 𝑜𝑜𝑜𝑜^

1 𝑎𝑎

1 2 b) i. 𝑥𝑥 = 2 ii. 𝑥𝑥 = − 2 iii. 𝑥𝑥 = 3 iv. 𝑥𝑥 = 5 v. Show that 16𝑎𝑎 2 𝑏𝑏 3 3𝑎𝑎 3 𝑏𝑏 ÷^

8𝑏𝑏 2 𝑎𝑎 9𝑎𝑎 3 𝑏𝑏 5 = 6𝑎𝑎𝑏𝑏^

5 16𝑎𝑎^2 𝑏𝑏^3 3𝑎𝑎 3 𝑏𝑏 ×^

9𝑎𝑎 3 𝑏𝑏 5 8𝑏𝑏 2 𝑎𝑎

= 2𝑎𝑎^

(^5) 𝑏𝑏 8 × 3 𝑏𝑏 3 𝑎𝑎 4 =^

2𝑎𝑎 1 𝑏𝑏 5 × 3 1 = 6𝑎𝑎𝑏𝑏^

5

Q2. Scientific Notation a) 4.5 x 10^2 b) 9.0 x 10^7 c) 3.5 x 10^0 d) 9.75 x 10-

e) 375 f) 39. g) 0. h) 0.

Q3. Calculations with scientific notation

a) 1.5 × 10− b) 3.375 × 10^9 c) −8.214 × 10^14 d) 8.8612 × 10^10

e) 3.076 × 10− f) 2.126 × 10− g) 500 s