base This is easy to see when each power is written in expanded - PDF document

n u m e r a c y n u m e r a c y Working with Exponents MPM1D: Principles of Mathematics Recap Simplify, then evaluate, 2 4 × 2 6 . Perform exponentiation before multiplication. 2 4 × 2 6 = 16 × 64 Exponent Laws (Numerical Bases) = 1 024 J. Garvin Note that 1 024 = 2 10 and that 2 4+6 = 2 10 . J. Garvin — Exponent Laws (Numerical Bases) Slide 1/18 Slide 2/18 n u m e r a c y n u m e r a c y Exponent Laws Exponent Laws In the earlier example, 2 4 × 2 6 , both powers have the same An expression involving some value (or base ) raised to some exponent is called a power . base. Their product is also a power of 2, and its exponent is the 2 3 ← exponent sum of the two given exponents. base → ���� This is easy to see when each power is written in expanded power form. 2 4 2 6 � �� � � �� � The resulting value is often referred to as a “power of” the 2 4 × 2 6 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 base. For example, 2 3 = 8, so we might say that 8 is a power � �� � 10 times of 2. = 2 10 Some people also use the term “power” to refer to the exponent, e.g. “2 to the power of 3”, but it is probably more accurate to use the phrase “2 to the exponent 3” instead. J. Garvin — Exponent Laws (Numerical Bases) J. Garvin — Exponent Laws (Numerical Bases) Slide 3/18 Slide 4/18 n u m e r a c y n u m e r a c y Exponent Laws Exponent Laws Example This property is true for the product of any two powers with the same base. Simplify, then evaluate, 3 2 × 3 3 . Product Rule for Exponents Since there is a common base of 3, add the exponents. For any two powers with the same base, their product is a power with the same base and an exponent equal to the sum 3 2 × 3 3 = 3 2+3 of the given exponents. = 3 5 This allows us to simplify expressions involving multiple = 243 powers with the same base. J. Garvin — Exponent Laws (Numerical Bases) J. Garvin — Exponent Laws (Numerical Bases) Slide 5/18 Slide 6/18

n u m e r a c y n u m e r a c y Exponent Laws Exponent Laws Example A similar rule exists when powers with like bases are divided. Consider the expression 3 7 Simplify, then evaluate, 2 8 × 2 10 . 3 5 . In expanded form, this is Since there is a common base of 2, add the exponents. 3 × 3 × 3 × 3 × 3 × 3 × 3 2 8 × 2 10 = 2 8+10 3 × 3 × 3 × 3 × 3 = 2 18 We can cancel each 3 in the denominator with a matching 3 = 262 144 in the numerator. 3 × 3 × ✁ 3 × ✁ 3 × ✁ 3 × ✁ 3 × ✁ 3 3 × ✁ 3 × ✁ 3 × ✁ 3 × ✁ ✁ 3 This leaves us with 3 × 3, or 3 2 . J. Garvin — Exponent Laws (Numerical Bases) J. Garvin — Exponent Laws (Numerical Bases) Slide 7/18 Slide 8/18 n u m e r a c y n u m e r a c y Exponent Laws Exponent Laws Note that 3 7 3 5 = 3 2 , and that 3 7 − 5 = 3 2 . Example Simplify, then evaluate, 10 9 10 6 . Quotient Rule for Exponents For any two powers with the same base, their quotient is a Since there is a common base of 10, subtract the exponents. power with the same base and an exponent equal to the difference of the given exponents. 10 9 10 6 = 10 9 − 6 It is possible to obtain negative exponents using this property. = 10 3 We will talk about these in more detail in later courses. = 1 000 J. Garvin — Exponent Laws (Numerical Bases) J. Garvin — Exponent Laws (Numerical Bases) Slide 9/18 Slide 10/18 n u m e r a c y n u m e r a c y Exponent Laws Exponent Laws � 5 2 � 3 ? Example What about the expression 5 2 � 3 = 5 2 × 5 2 × 5 2 . Simplify, then evaluate, 15 4 ÷ 15 3 . � Expanding this, we get Using the product rule, we can add the exponents, since Since there is a common base of 15, subtract the exponents. there is a common base of 5. 15 4 ÷ 15 3 = 15 4 − 3 5 2 × 5 2 × 5 2 = 5 2+2+2 = 15 1 = 5 6 = 15 5 2 � 3 = 5 6 and that 5 2 × 3 = 5 6 . � Note that J. Garvin — Exponent Laws (Numerical Bases) J. Garvin — Exponent Laws (Numerical Bases) Slide 11/18 Slide 12/18

n u m e r a c y n u m e r a c y Exponent Laws Exponent Laws This is a third exponent law, applying to a “power of a Example power”. � 2 5 � 3 . Simplify, then evaluate, Power of a Power Rule for Exponents Multiply the exponents. For any power raised to an exponent, its value is a power with the same base and an exponent equal to the product of 2 5 � 3 = 2 5 × 3 � the given exponents. = 2 15 There are other exponent laws in addition to these three, but = 32 768 they will be covered in later courses. J. Garvin — Exponent Laws (Numerical Bases) J. Garvin — Exponent Laws (Numerical Bases) Slide 13/18 Slide 14/18 n u m e r a c y n u m e r a c y Exponent Laws Exponent Laws Sometimes, expressions involve multiple exponent laws. Example � 2 6 × 2 3 � 2 . They can usually be applied in any order, but try to keep Simplify things simple – make values smaller, rather than larger, and try to cancel things as early as possible. Add the exponents inside of the brackets first, then multiply. 2 6 × 2 3 � 2 = � � 2 6+3 � 2 � 2 9 � 2 = = 2 9 × 2 = 2 18 J. Garvin — Exponent Laws (Numerical Bases) J. Garvin — Exponent Laws (Numerical Bases) Slide 15/18 Slide 16/18 n u m e r a c y n u m e r a c y Exponent Laws Questions? Example � 4 � 5 13 Simplify . 5 11 Subtract the exponents inside of the brackets first, then multiply. � 5 13 � 4 5 13 − 11 � 4 � = 5 11 � 5 2 � 4 = = 5 2 × 4 = 5 8 J. Garvin — Exponent Laws (Numerical Bases) J. Garvin — Exponent Laws (Numerical Bases) Slide 17/18 Slide 18/18