Exponents and scientific notation

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1. Exponent rules: products, quotients, powers of powers

Warm-up

Three quick questions from earlier lessons.

Discover

23⋅242^3 \cdot 2^4 means (2⋅2⋅2)(2⋅2⋅2⋅2)(2 \cdot 2 \cdot 2)(2 \cdot 2 \cdot 2 \cdot 2). How many factors of 2 is that?

  1. 7
  2. 12

Key idea

Exponent rules come from counting factors.

For the same base: am⋅an=am+na^m \cdot a^n = a^{m+n} (add), am÷an=am−na^m \div a^n = a^{m-n} (subtract), (am)n=amn(a^m)^n = a^{mn} (multiply). Also an⋅bn=(ab)na^n \cdot b^n = (ab)^n.

Worked example

Simplify (52)3⋅5457\dfrac{(5^2)^3 \cdot 5^4}{5^7}.

(52)3=56(5^2)^3 = 5^6
Power of a power.
56⋅54=5105^6 \cdot 5^4 = 5^{10}
Product.
510÷57=535^{10} \div 5^7 = 5^3
Quotient.

Your turn, guided

Your turn: 35⋅32=3?3^5 \cdot 3^2 = 3^{?}.

77
5 + 2.

Practice ★

  1. 62⋅65=6?6^{ 2 } \cdot 6^{ 5 } = 6^{?}. What is the exponent?
  2. 85⋅89=8?8^{ 5 } \cdot 8^{ 9 } = 8^{?}. What is the exponent?
  1. 312÷35=3?3^{ 12 } \div 3^{ 5 } = 3^{?}. What is the exponent?

Watch out!

Practice ★★

  1. (94)4=9?\left(9^{ 4 }\right)^{ 4 } = 9^{?}. What is the exponent?
  2. (63)3=6?\left(6^{ 3 }\right)^{ 3 } = 6^{?}. What is the exponent?

Exit ticket

Three questions to check you have got it.

  1. 62⋅65=6?6^{ 2 } \cdot 6^{ 5 } = 6^{?}. What is the exponent?
  2. 85⋅89=8?8^{ 5 } \cdot 8^{ 9 } = 8^{?}. What is the exponent?
  3. 93⋅97=9?9^{ 3 } \cdot 9^{ 7 } = 9^{?}. What is the exponent?

Summary card

Exponent rules

Multiply: add. Divide: subtract. Power of a power: multiply.

Watch out: count the factors when in doubt.

2. Zero and negative exponents

Warm-up

Three quick questions from earlier lessons.

Discover

103=100010^3 = 1000, 102=10010^2 = 100, 101=1010^1 = 10. Each step divides by 10. What comes next, 10010^0?

  1. 11
  2. 00

Key idea

Keep dividing to go below exponent 1.

For a≠0a \neq 0: a0=1a^0 = 1 and a−n=1ana^{-n} = \dfrac{1}{a^n}. The exponent rules still work with zero and negative exponents.

Worked example

2−32^{-3} and 105⋅10−210^5 \cdot 10^{-2}.

2−3=182^{-3} = \tfrac{1}{8}
One over 2³.
105⋅10−2=10310^5 \cdot 10^{-2} = 10^3
5 + (−2).

Your turn, guided

Your turn: 10−210^{-2} as a fraction.

1100\tfrac{1}{100}
1/10².

Practice ★

  1. Write 5−15^{-1} as a fraction.
  2. Write 4−14^{-1} as a fraction.
  1. What is 51051^0?

Watch out!

Practice ★★

  1. 106⋅10−9=10?10^{ 6 } \cdot 10^{-9} = 10^{?}. What is the exponent?
  2. 108⋅10−9=10?10^{ 8 } \cdot 10^{-9} = 10^{?}. What is the exponent?

Exit ticket

Three questions to check you have got it.

  1. Write 5−15^{-1} as a fraction.
  2. Write 4−14^{-1} as a fraction.
  3. Write 2−12^{-1} as a fraction.

Summary card

Zero and negative exponents

a0=1a^0 = 1; a−n=1/ana^{-n} = 1/a^n.

Watch out: negative exponent, small positive number.

3. Scientific notation

Warm-up

Three quick questions from earlier lessons.

Discover

The Sun is about 150,000,000 km from Earth. Which is the same number?

  1. 1.5×1081.5 \times 10^8
  2. 1.5×1091.5 \times 10^9

Key idea

Scientific notation: one digit before the point, times a power of 10.

A number is in scientific notation when it is written a×10na \times 10^n with 1≤a<101 \le a < 10 and nn an integer. Large numbers have positive nn; numbers between 0 and 1 have negative nn. Comparing: first compare the powers of 10.

Worked example

0.000420.00042.

4.24.2
Move the point 4 places right.
4.2×10−44.2 \times 10^{-4}
Small number: negative exponent.

Your turn, guided

Your turn: 6,000,000=6×10?6{,}000{,}000 = 6 \times 10^{?}.

66
Six zeros.

Practice ★

  1. Write 900000000900000000 in scientific notation as 9×10?9 \times 10^{?}. What is the exponent?
  2. Write 80000008000000 in scientific notation as 8×10?8 \times 10^{?}. What is the exponent?
  1. How many times as large as 8×1048 \times 10^{ 4 } is 8×1078 \times 10^{ 7 }?

Watch out!

Practice ★★

  1. When 2×10−62 \times 10^{-6} is written as a decimal, how many zeros are there between the decimal point and the digit 2?
  2. When 6×10−86 \times 10^{-8} is written as a decimal, how many zeros are there between the decimal point and the digit 6?

Exit ticket

Three questions to check you have got it.

  1. Write 900000000900000000 in scientific notation as 9×10?9 \times 10^{?}. What is the exponent?
  2. Write 80000008000000 in scientific notation as 8×10?8 \times 10^{?}. What is the exponent?
  3. Write 800000000800000000 in scientific notation as 8×10?8 \times 10^{?}. What is the exponent?

Summary card

Scientific notation

a×10na \times 10^n with 1≤a<101 \le a < 10.

Watch out: count the places the point moves, not the digits.

4. Computing with scientific notation

Warm-up

Three quick questions from earlier lessons.

Discover

(2×103)(4×105)(2 \times 10^3)(4 \times 10^5). Group the numbers and the powers of ten. What do you get?

  1. 8×1088 \times 10^8
  2. 8×10158 \times 10^{15}

Key idea

Compute with the parts separately.

Multiply or divide: work on the numbers and the powers of 10 separately, then adjust to scientific notation. Add or subtract: rewrite with the same power of 10 first, then add the numbers; the power of 10 stays.

Worked example

(9×107)÷(3×102)(9 \times 10^7) \div (3 \times 10^2) and 5×104+7×1045 \times 10^4 + 7 \times 10^4.

3×1053 \times 10^5
9 ÷ 3, 7 − 2.
12×104=1.2×10512 \times 10^4 = 1.2 \times 10^5
Like terms.

Your turn, guided

Your turn: (3×102)(2×106)=6×10?(3 \times 10^2)(2 \times 10^6) = 6 \times 10^{?}.

88
2 + 6.

Practice ★

  1. Compute (3×104)⋅(2×102)(3 \times 10^{ 4 }) \cdot (2 \times 10^{ 2 }). (An expression with a power of 10 is fine.)
  2. Compute (3×103)⋅(4×103)(3 \times 10^{ 3 }) \cdot (4 \times 10^{ 3 }). (An expression with a power of 10 is fine.)
  1. Compute (4×109)÷(2×104)(4 \times 10^{ 9 }) \div (2 \times 10^{ 4 }). (An expression with a power of 10 is fine.)

Watch out!

Practice ★★

  1. Compute 2×104+2×1042 \times 10^{ 4 } + 2 \times 10^{ 4 }. (An expression with a power of 10 is fine.)
  2. Compute 3×104+1×1043 \times 10^{ 4 } + 1 \times 10^{ 4 }. (An expression with a power of 10 is fine.)
  1. Boss problem. A phone does about 3×1093 \times 10^9 operations per second. An old computer did about 1×1041 \times 10^4 operations per second.

Exit ticket

Three questions to check you have got it.

  1. Compute (3×104)⋅(2×102)(3 \times 10^{ 4 }) \cdot (2 \times 10^{ 2 }). (An expression with a power of 10 is fine.)
  2. Compute (3×103)⋅(4×103)(3 \times 10^{ 3 }) \cdot (4 \times 10^{ 3 }). (An expression with a power of 10 is fine.)
  3. Compute (2×102)⋅(2×103)(2 \times 10^{ 2 }) \cdot (2 \times 10^{ 3 }). (An expression with a power of 10 is fine.)

Summary card

Computing in scientific notation

Multiply/divide: separately. Add/subtract: same power first.

Watch out: adding does not add the exponents.

Answers: Exponents and scientific notation

1. Exponent rules: products, quotients, powers of powers

A

  1. 77
  2. 1414

B

  1. 77

C

  1. 1616
  2. 99

D

  1. 77
  2. 1414
  3. 1010

2. Zero and negative exponents

A

  1. 15\frac{1}{5}
  2. 14\frac{1}{4}

B

  1. 11

C

  1. −3-3
  2. −1-1

D

  1. 15\frac{1}{5}
  2. 14\frac{1}{4}
  3. 12\frac{1}{2}

3. Scientific notation

A

  1. 88
  2. 66

B

  1. 10001000

C

  1. 55
  2. 77

D

  1. 88
  2. 66
  3. 88

4. Computing with scientific notation

A

  1. 60000006000000
  2. 1200000012000000

B

  1. 200000200000

C

  1. 4000040000
  2. 4000040000

D

  1. 30000003000000

E

  1. 60000006000000
  2. 1200000012000000
  3. 400000400000