Exponents
Exponents
Introduction RECALL: Exponents provide a shorter way of representing the products of repeated numbers.
Base
n a
Exponent
a×a×a×a×a×…×a= n factors of a.
n a
Exponents with Signed Numbers If the base is negative and the power is even, the result is a positive number.
2 (–5)
= (–5) × (–5) = 25
4 (–3)
= (–3) × (–3) × (–3) × (–3) =9×9 = 81
Exponents with Signed Numbers If the base is negative and the power is odd, the result is a negative number.
5 (–3)
= (–3)×(–3)×(–3)×(–3)×(–3) = 9 × 9 × (–3) = 81 × (–3) = –243
Exponents with Signed Numbers Be careful when identifying the base of a signed number!
4 (–2)
= (–2) × (–2) × (–2) × (–2) = 16
–24 = (–1) × 2 × 2 × 2 × 2 = –16
Rules of Exponents When multiplying powers of the same bases, add the exponents.
Product Rule
m n (a )(a )
=
m+n a
1+1+1+1+1 or 3+2
5 = 2 × 2 × 2 × 2 × 2 × 2
3 2
2 2
We have five repeated multiplications of 2.
Rules of Exponents When dividing powers with the same bases, subtract the exponent of the denominator from that of the numerator. Quotient Rule
5 x 3 x
m a n a
=
=
5−3 x
m−n a
=
2 x
Evaluate the following expression: 34 × 33 ÷ 32 a. b. c. d.
81 243 729 2187
Evaluate the following expression: 34 × 33 ÷ 3 2
Rules of Exponents If any product of numbers ‘a’ and ‘b’ is raised to a power of n, then each number in the product is also raised to the same power.
Power of a Product
n (a×b)
(4 ×
2 3)
=
n n a ×b
=
2 4
×
2 3
Rules of Exponents If any quotient with numerator ‘a’ and a non-zero denominator ‘b’ is raised to a power ‘n’, then the numerator ‘a’ and denominator ‘b’ are both raised to the same power. Power of a Quotient
n a n b
a n = b 3 3 3 = 3 3 8 8
( ) ( )
Rules of Exponents To raise a power to a power, multiply the exponents.
Power of a Power
m n (a )
mn a
=
3 2 (2 )
3×2 2
= = 26 (2 × 2 × 2) × ( 2 × 2 × 2 ) We have six repeated multiplications of 2.
Rules of Exponents 1 or 0 as an exponent
1 x = 1 5 =
x
0 x
= 1
5
0 5
= 1
Which of the following is not equal to 1? a. b. c. d.
10 11 (–25x3 + y)0 –20
Which of the following is not equal to 1?
Rules of Exponents Using the quotient rule: Why does x0 = 1?
𝑥𝑛 𝑛−𝑛 0 = 𝑥 = 𝑥 𝑥𝑛 However, using our knowledge of fractions:
𝑥𝑛 = 1, 𝑛 𝑥
𝑤ℎ𝑒𝑟𝑒 𝑥 ≠ 0
Therefore:
𝑥 0 = 1,
𝑤ℎ𝑒𝑟𝑒 𝑥 ≠ 0
Rules of Exponents A base raised to a negative power is equal to the reciprocal of the base raised to the absolute value of the exponent.
Negative Exponent
-n a −𝟒
𝟐(𝒙 + 𝒚)
1 = n a
𝟐 = (𝒙 + 𝒚)𝟒
𝟏𝟎 𝟓 = 𝟏𝟎𝒛 𝒛−𝟓
Rules of Exponents Using the product and 0 as an exponent rules: Why does 𝟏 𝒙−𝒏 = 𝒙𝒏 ?
𝑥 𝑛 × 𝑥 −𝑛 = 𝑥 𝑛+(−𝑛) = 𝑥 0 = 1 However, using our knowledge of fractions:
1 𝑥𝑛 × 𝑛 = 1 𝑥 Therefore:
𝑥
−𝑛
1 = 𝑛 𝑥
Rules of Exponents
Reminder: Be careful identifying the base when dealing with negative exponents!
2𝑥 𝒙 –4 5𝑦 𝒚 –3
=
–4 (2𝑥) 𝟐𝒙
(5𝑦) 𝟓𝒚 –3
𝒚 –1 𝑥3𝑦 𝑥 –1 𝑦 3 𝒙
2𝑦 3 5𝑥 4
= =
(5𝑦)3 (2𝑥)4 3 1 𝑥 𝑥
=
125𝑦 3 16𝑥 4
𝑥4 = 4 3 1 𝑦 𝑦 𝑦
Simplifying Exponential Expressions To simplify an exponential expression:
1. Remove all negative exponents.
2. Reduce fractions to simplest terms. 3. Perform all arithmetic.
Simplify the following expression: (𝟒𝒂𝟓 )𝟑 (𝟏𝟐𝒂𝒃)𝟎 (𝟐𝒂𝒃−𝟏 )−𝟐
Simplify the following expression: −𝟑𝒙−𝟑 (𝟏𝟖𝒙𝟐 )−𝟏 𝟏 −𝟓𝟒𝒙𝟓 −𝟏 b. 𝟔𝒙𝟓
a.
c.
𝟏 −𝟓𝟒𝒙𝟔
−𝟏 d. 𝟔𝒙𝟔
Simplify the following expression: −𝟑𝒙−𝟑 (𝟏𝟖𝒙𝟐 )−𝟏
Introduction RECALL: Exponents provide a shorter way of representing the products of repeated numbers.
Base
n a
Exponent
a×a×a×a×a×…×a= n factors of a.
n a
Exponents with Signed Numbers If the base is negative and the power is even, the result is a positive number.
2 (–5)
= (–5) × (–5) = 25
4 (–3)
= (–3) × (–3) × (–3) × (–3) =9×9 = 81
Exponents with Signed Numbers If the base is negative and the power is odd, the result is a negative number.
5 (–3)
= (–3)×(–3)×(–3)×(–3)×(–3) = 9 × 9 × (–3) = 81 × (–3) = –243
Exponents with Signed Numbers Be careful when identifying the base of a signed number!
4 (–2)
= (–2) × (–2) × (–2) × (–2) = 16
–24 = (–1) × 2 × 2 × 2 × 2 = –16
Rules of Exponents When multiplying powers of the same bases, add the exponents.
Product Rule
m n (a )(a )
=
m+n a
1+1+1+1+1 or 3+2
5 = 2 × 2 × 2 × 2 × 2 × 2
3 2
2 2
We have five repeated multiplications of 2.
Rules of Exponents When dividing powers with the same bases, subtract the exponent of the denominator from that of the numerator. Quotient Rule
5 x 3 x
m a n a
=
=
5−3 x
m−n a
=
2 x
Evaluate the following expression: 34 × 33 ÷ 32 a. b. c. d.
81 243 729 2187
Evaluate the following expression: 34 × 33 ÷ 3 2
Rules of Exponents If any product of numbers ‘a’ and ‘b’ is raised to a power of n, then each number in the product is also raised to the same power.
Power of a Product
n (a×b)
(4 ×
2 3)
=
n n a ×b
=
2 4
×
2 3
Rules of Exponents If any quotient with numerator ‘a’ and a non-zero denominator ‘b’ is raised to a power ‘n’, then the numerator ‘a’ and denominator ‘b’ are both raised to the same power. Power of a Quotient
n a n b
a n = b 3 3 3 = 3 3 8 8
( ) ( )
Rules of Exponents To raise a power to a power, multiply the exponents.
Power of a Power
m n (a )
mn a
=
3 2 (2 )
3×2 2
= = 26 (2 × 2 × 2) × ( 2 × 2 × 2 ) We have six repeated multiplications of 2.
Rules of Exponents 1 or 0 as an exponent
1 x = 1 5 =
x
0 x
= 1
5
0 5
= 1
Which of the following is not equal to 1? a. b. c. d.
10 11 (–25x3 + y)0 –20
Which of the following is not equal to 1?
Rules of Exponents Using the quotient rule: Why does x0 = 1?
𝑥𝑛 𝑛−𝑛 0 = 𝑥 = 𝑥 𝑥𝑛 However, using our knowledge of fractions:
𝑥𝑛 = 1, 𝑛 𝑥
𝑤ℎ𝑒𝑟𝑒 𝑥 ≠ 0
Therefore:
𝑥 0 = 1,
𝑤ℎ𝑒𝑟𝑒 𝑥 ≠ 0
Rules of Exponents A base raised to a negative power is equal to the reciprocal of the base raised to the absolute value of the exponent.
Negative Exponent
-n a −𝟒
𝟐(𝒙 + 𝒚)
1 = n a
𝟐 = (𝒙 + 𝒚)𝟒
𝟏𝟎 𝟓 = 𝟏𝟎𝒛 𝒛−𝟓
Rules of Exponents Using the product and 0 as an exponent rules: Why does 𝟏 𝒙−𝒏 = 𝒙𝒏 ?
𝑥 𝑛 × 𝑥 −𝑛 = 𝑥 𝑛+(−𝑛) = 𝑥 0 = 1 However, using our knowledge of fractions:
1 𝑥𝑛 × 𝑛 = 1 𝑥 Therefore:
𝑥
−𝑛
1 = 𝑛 𝑥
Rules of Exponents
Reminder: Be careful identifying the base when dealing with negative exponents!
2𝑥 𝒙 –4 5𝑦 𝒚 –3
=
–4 (2𝑥) 𝟐𝒙
(5𝑦) 𝟓𝒚 –3
𝒚 –1 𝑥3𝑦 𝑥 –1 𝑦 3 𝒙
2𝑦 3 5𝑥 4
= =
(5𝑦)3 (2𝑥)4 3 1 𝑥 𝑥
=
125𝑦 3 16𝑥 4
𝑥4 = 4 3 1 𝑦 𝑦 𝑦
Simplifying Exponential Expressions To simplify an exponential expression:
1. Remove all negative exponents.
2. Reduce fractions to simplest terms. 3. Perform all arithmetic.
Simplify the following expression: (𝟒𝒂𝟓 )𝟑 (𝟏𝟐𝒂𝒃)𝟎 (𝟐𝒂𝒃−𝟏 )−𝟐
Simplify the following expression: −𝟑𝒙−𝟑 (𝟏𝟖𝒙𝟐 )−𝟏 𝟏 −𝟓𝟒𝒙𝟓 −𝟏 b. 𝟔𝒙𝟓
a.
c.
𝟏 −𝟓𝟒𝒙𝟔
−𝟏 d. 𝟔𝒙𝟔
Simplify the following expression: −𝟑𝒙−𝟑 (𝟏𝟖𝒙𝟐 )−𝟏
