Exponentials - Grade 10 [CAPS] *
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1 OpenStx-CNX module: m859 Exponentils - Grde 0 [CAPS] * Free High School Science Texts Project Bsed on Exponentils by Rory Adms Free High School Science Texts Project Mrk Horner Hether Willims This work is produced by OpenStx-CNX nd licensed under the Cretive Commons Attribution License.0 Introduction In this chpter, you will lern bout the short cuts to writing. This is known s writing number in exponentil nottion. Denition Exponentil nottion is short wy of writing the sme number multiplied by itself mny times. insted of 5 5 5, we write 5 to show tht the number 5 is multiplied by itself times nd we sy 5 to the power of. Likewise 5 is 5 5 nd 5 is. We will now hve closer look t writing numbers using exponentil nottion. Denition : Exponentil Nottion Exponentil nottion mens number written like where n is n integer nd cn be ny rel number. exponent or index. The n th power of is dened s: n is clled the bse nd n is clled the with multiplied by itself n times. n (n times) * Version.: Jun, 0 5:0 pm
2 OpenStx-CNX module: m859 We cn lso dene wht it mens if we hve negtive exponent n. Then, n (n times) tip: Exponentils If n is n even integer, then n will lwys be positive for ny non-zero rel number. lthough is negtive, ( ) 4 is positive nd so is ( ) 4. Khn Acdemy video on Exponents - This medi object is Flsh object. Plese view or downlod it t < Figure Khn Acdemy video on Exponents- This medi object is Flsh object. Plese view or downlod it t < Figure Lws of Exponents There re severl lws we cn use to mke working with exponentil numbers esier. Some of these lws might hve been seen in erlier grdes, but we will list ll the lws here for esy reference nd explin ech lw in detil, so tht you cn understnd them nd not only remember them. 0 m n m+n n n m n m n (b) n n b n ( m ) n mn. Exponentil Lw : 0 Our denition of exponentil nottion shows tht 0, ( 0)
3 OpenStx-CNX module: m859 To convince yourself of why this is true, use the fourth exponentil lw bove (division of exponents) nd consider wht hppens when m n. x 0 nd ( ) 0... Appliction using Exponentil Lw : 0, ( 0) (6 + ) 0 4. ( 6) Click here for the solution. Exponentil Lw : m n m+n Khn Acdemy video on Exponents - This medi object is Flsh object. Plese view or downlod it t < Figure Our denition of exponentil nottion shows tht m n... (m times)... (n times)... (m + n times) m+n 7 ( ) ( ) 7+ 0 note: This simple lw is the reson why exponentils were originlly invented. In the dys before clcultors, ll multipliction hd to be done by hnd with pencil nd pd of pper. Multipliction tkes very long time to do nd is very tedious. Adding numbers however, is very esy nd quick to do. If you look t wht this lw is sying you will relise tht it mens tht dding the exponents of two exponentil numbers (of the sme bse) is the sme s multiplying the two numbers together. This ment tht for certin numbers, there ws no need to ctully multiply the numbers together in order to nd out wht their multiple ws. This sved mthemticins lot of time, which they could use to do something more productive.
4 OpenStx-CNX module: m Appliction using Exponentil Lw : m n m+n. x x 5. 4 [Tke note tht the bse () stys the sme.]. Click here for the solution. Exponentil Lw : n n, 0 Our denition of exponentil nottion for negtive exponent shows tht n... (n times) n (n times) This mens tht minus sign in the exponent is just nother wy of showing tht the whole exponentil number is to be divided insted of multiplied. 7 7 This lw is useful in helping us simplify frctions where there re exponents in both the denomintor nd the numertor. For exmple: Appliction using Exponentil Lw : n n, 0.. ). ( m 4. n 4 5. x 4 5 x Click here for the solution.4 Exponentil Lw 4: m n m n We lredy relised with lw tht minus sign is nother wy of sying tht the exponentil number is to be divided insted of multiplied. Lw 4 is just more generl wy of sying the sme thing. We get this lw by multiplying lw by m on both sides nd using lw. m n m n m n
5 OpenStx-CNX module: m Khn cdemy video on exponents - 4 This medi object is Flsh object. Plese view or downlod it t < Figure.4. Appliction using Exponentil Lw 4: m n m n x 4 Click here for the solution 4.5 Exponentil Lw 5: (b) n n b n The order in which two rel numbers re multiplied together does not mtter. Therefore, (b) n b b... b (n times)... (n times) b b... b (n times) n b n ( ) 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 ( 4)
6 OpenStx-CNX module: m Appliction using Exponentil Lw 5: (b) n n b n. (xy) x y. ( ) 7 b. (5) Click here for the solution 5.6 Exponentil Lw 6: ( m ) n mn We cn nd the exponentil of n exponentil of number. An exponentil of number is just rel number. So, even though the sentence sounds complicted, it is just sying tht you cn nd the exponentil of number nd then tke the exponentil of tht number. You just tke the exponentil twice, using the nswer of the rst exponentil s the rgument for the second one. ( m ) n m m... m (n times)... (m n times) mn ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 6 ( ).6. Appliction using Exponentil Lw 6: ( m ) n mn. ( x ) 4 [ ( 4. ) ]. ( n+) Click here for the solution 6 Exercise : Simplifying indices (Solution on p. 9.) x 5 x 9 Simplify: 5 x.6. Investigtion : Exponentil Numbers Mtch the nswers to the questions, by lling in the correct nswer into the Answer column. Possible nswers re:,,,, 8. Answers my be repeted
7 OpenStx-CNX module: m859 7 Question 7 ( ) Answer ( ) ( ) Tble We will use ll these lws in Equtions nd Inequlities to help us solve exponentil equtions. The following video gives n exmple on using some of the concepts covered in this chpter. Khn Acdemy video on Exponents - 5 This medi object is Flsh object. Plese view or downlod it t < Figure 4 Summry ˆ Exponentil nottion mens number written like n where n is n integer nd cn be ny rel number. ˆ is clled the bse nd n is clled the exponent or index. ˆ The n th power of is dened s: n (n times) ˆ There re six lws of exponents: Exponentil Lw : 0 Exponentil Lw : m n m+n Exponentil Lw : n, 0 n Exponentil Lw 4: m n m n Exponentil Lw 5: (b) n n b n Exponentil Lw 6: ( m ) n mn 5 End of Chpter Exercises. Simplify s fr s possible:. 0 0 b. 0 c. (xyz) 0 [ (x d. 4 y 7 z ) 5( 5x 9 y z 4) ] 0 e. (x) f. ( x)
8 OpenStx-CNX module: m859 8 g. (x) 4 h. ( x) 4 Click here for the solution 7. Simplify without using clcultor. Leve your nswers with positive exponents.. x (x) b. 5x ( ) x 5 c. b 5 b+ Click here for the solution 8. Simplify, showing ll steps: b. m+n+p m+n+p m n n 9 c. 7 n d. ( x y b ) x+ x 8 e. 4 x x x x x f. 6 Click here for the solution 9 4. Simplify, without using clcultor:. ( ) ( ) ( ) 4 b. ( + ) n 9 n 7 c. 8 n n n+ 8 d. 4 n Click here for the solution
9 OpenStx-CNX module: m859 9 Solutions to Exercises in this Module Solution to Exercise (p. 6) Step. Step. Step. 5x ( ) x (5.) x x 4 5 x 5 x x 5 x x+ x 4 x+ 5 5
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