Math 71B 7.2 – Rational Exponents
Let’s define what it means to have a rational # as an exponent: 𝒂 𝟏 𝒏 = 𝒏 𝒂 Ex 1. 8 1 1 2 = 81 =𝟗 −64 1/3 = 3 −64 =−𝟒 7 𝑥 2 𝑦 5𝑧 = 𝒙 𝟐 𝒚 𝟓𝒛 𝟏 𝟕 (we’ll talk about how to simplify this further later on…)
Let’s define what it means to have a rational # as an exponent: 𝒂 𝟏 𝒏 = 𝒏 𝒂 Ex 1. 8 1 1 2 = 81 =𝟗 −64 1/3 = 3 −64 =−𝟒 7 𝑥 2 𝑦 5𝑧 = 𝒙 𝟐 𝒚 𝟓𝒛 𝟏 𝟕 (we’ll talk about how to simplify this further later on…)
Let’s define what it means to have a rational # as an exponent: 𝒂 𝟏 𝒏 = 𝒏 𝒂 Ex 1. 8 1 1 2 = 81 =𝟗 −64 1/3 = 3 −64 =−𝟒 7 𝑥 2 𝑦 5𝑧 = 𝒙 𝟐 𝒚 𝟓𝒛 𝟏 𝟕 (we’ll talk about how to simplify this further later on…)
Let’s define what it means to have a rational # as an exponent: 𝒂 𝟏 𝒏 = 𝒏 𝒂 Ex 1. 8 1 1 2 = 81 =𝟗 −64 1/3 = 3 −64 =−𝟒 7 𝑥 2 𝑦 5𝑧 = 𝒙 𝟐 𝒚 𝟓𝒛 𝟏 𝟕 (we’ll talk about how to simplify this further later on…)
Let’s define what it means to have a rational # as an exponent: 𝒂 𝟏 𝒏 = 𝒏 𝒂 Ex 1. 8 1 1 2 = 81 =𝟗 −64 1/3 = 3 −64 =−𝟒 7 𝑥 2 𝑦 5𝑧 = 𝒙 𝟐 𝒚 𝟓𝒛 𝟏 𝟕 (we’ll talk about how to simplify this further later on…)
Let’s define what it means to have a rational # as an exponent: 𝒂 𝟏 𝒏 = 𝒏 𝒂 Ex 1. 8 1 1 2 = 81 =𝟗 −64 1/3 = 3 −64 =−𝟒 7 𝑥 2 𝑦 5𝑧 = 𝒙 𝟐 𝒚 𝟓𝒛 𝟏 𝟕 (we’ll talk about how to simplify this further later on…)
Note: We can write 𝑎 2 3 in a couple of different ways: 𝑎 2 3 = 𝑎 1 3 2 = 𝟑 𝒂 𝟐 or 𝑎 2 3 = 𝑎 2 1 3 = 𝟑 𝒂 𝟐
Note: We can write 𝑎 2 3 in a couple of different ways: 𝑎 2 3 = 𝑎 1 3 2 = 𝟑 𝒂 𝟐 or 𝑎 2 3 = 𝑎 2 1 3 = 𝟑 𝒂 𝟐
Ex 2. Use radical notation to rewrite each expression and simplify
Ex 2. Use radical notation to rewrite each expression and simplify
Ex 2. Use radical notation to rewrite each expression and simplify
Ex 2. Use radical notation to rewrite each expression and simplify
Ex 2. Use radical notation to rewrite each expression and simplify
Ex 2. Use radical notation to rewrite each expression and simplify
Ex 3. Rewrite with rational exponents: 3 5 4 = 𝟓 𝟒 𝟑 4 11 𝑥 2 𝑦 9 = 𝟏𝟏 𝒙 𝟐 𝒚 𝟗 𝟒
Ex 3. Rewrite with rational exponents: 3 5 4 = 𝟓 𝟒 𝟑 4 11 𝑥 2 𝑦 9 = 𝟏𝟏 𝒙 𝟐 𝒚 𝟗 𝟒
Ex 3. Rewrite with rational exponents: 3 5 4 = 𝟓 𝟒 𝟑 4 11 𝑥 2 𝑦 9 = 𝟏𝟏 𝒙 𝟐 𝒚 𝟗 𝟒
Radical Notation Rational Exponent 7 𝑥 𝑥 3 𝑥 5 𝑥 3 𝑥 1 8 𝑥 1 2 𝑥 5 6 𝑥 7 2
Radical Notation Rational Exponent 7 𝑥 𝒙 𝟏 𝟕 𝑥 3 𝑥 5 𝑥 3 𝑥 1 8 𝑥 1 2 𝑥 5 6 𝑥 7 2
Radical Notation Rational Exponent 7 𝑥 𝒙 𝟏 𝟕 𝑥 𝒙 𝟏 𝟐 3 𝑥 5 𝑥 3 𝑥 1 8 𝑥 1 2 𝑥 5 6 𝑥 7 2
Radical Notation Rational Exponent 7 𝑥 𝒙 𝟏 𝟕 𝑥 𝒙 𝟏 𝟐 3 𝑥 5 𝒙 𝟓 𝟑 𝑥 3 𝑥 1 8 𝑥 1 2 𝑥 5 6 𝑥 7 2
Radical Notation Rational Exponent 7 𝑥 𝒙 𝟏 𝟕 𝑥 𝒙 𝟏 𝟐 3 𝑥 5 𝒙 𝟓 𝟑 𝑥 3 𝒙 𝟑 𝟐 𝑥 1 8 𝑥 1 2 𝑥 5 6 𝑥 7 2
Radical Notation Rational Exponent 7 𝑥 𝒙 𝟏 𝟕 𝑥 𝒙 𝟏 𝟐 3 𝑥 5 𝒙 𝟓 𝟑 𝑥 3 𝒙 𝟑 𝟐 𝟖 𝒙 𝑥 1 8 𝑥 1 2 𝑥 5 6 𝑥 7 2
Radical Notation Rational Exponent 7 𝑥 𝒙 𝟏 𝟕 𝑥 𝒙 𝟏 𝟐 3 𝑥 5 𝒙 𝟓 𝟑 𝑥 3 𝒙 𝟑 𝟐 𝟖 𝒙 𝑥 1 8 𝒙 𝑥 1 2 𝑥 5 6 𝑥 7 2
Radical Notation Rational Exponent 7 𝑥 𝒙 𝟏 𝟕 𝑥 𝒙 𝟏 𝟐 3 𝑥 5 𝒙 𝟓 𝟑 𝑥 3 𝒙 𝟑 𝟐 𝟖 𝒙 𝑥 1 8 𝒙 𝑥 1 2 𝟔 𝒙 𝟓 𝑥 5 6 𝑥 7 2
Radical Notation Rational Exponent 7 𝑥 𝒙 𝟏 𝟕 𝑥 𝒙 𝟏 𝟐 3 𝑥 5 𝒙 𝟓 𝟑 𝑥 3 𝒙 𝟑 𝟐 𝟖 𝒙 𝑥 1 8 𝒙 𝑥 1 2 𝟔 𝒙 𝟓 𝑥 5 6 𝒙 𝟕 𝑥 7 2
Ex 4. Rewrite with a positive exponent. Simplify if possible
Ex 4. Rewrite with a positive exponent. Simplify if possible
Ex 4. Rewrite with a positive exponent. Simplify if possible
Ex 4. Rewrite with a positive exponent. Simplify if possible
Ex 4. Rewrite with a positive exponent. Simplify if possible
Ex 4. Rewrite with a positive exponent. Simplify if possible
Ex 4. Rewrite with a positive exponent. Simplify if possible
Note: The same rules that you learned before apply when working with rational exponents. 𝑏 𝑚 ⋅ 𝑏 𝑛 = 𝑏 𝑚+𝑛 𝑏 𝑚 𝑏 𝑛 = 𝑏 𝑚−𝑛 𝑏 𝑚 𝑛 = 𝑏 𝑚𝑛 𝑎𝑏 𝑛 = 𝑎 𝑛 𝑏 𝑛 𝑎 𝑏 𝑛 = 𝑎 𝑛 𝑏 𝑛 𝑏 −𝑛 = 1 𝑏 𝑛 𝑏 0 =1 When is an expression with rational exponents simplified? Basically, when you’re done using the above rules. That means…
1. No parentheses around products or quotients 1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )
1. No parentheses around products or quotients 1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )
1. No parentheses around products or quotients 1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )
1. No parentheses around products or quotients 1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )
1. No parentheses around products or quotients 1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )
1. No parentheses around products or quotients 1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )
1. No parentheses around products or quotients 1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )
1. No parentheses around products or quotients 1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )
Ex 5. Simplify (assume all variables represent positive numbers): 7 1 2 ⋅ 7 1 3 = 50 𝑥 1 3 10 𝑥 4 3 = 𝑥 − 3 5 𝑦 1 4 1 3 = 5 𝑦 2 10 𝑦 3 = 𝑥 ⋅ 3 𝑥 = 3 𝑥 = 6 𝑎 𝑏 2 ⋅ 3 𝑎 2 𝑏 =