Lesso : Workig with the Nuber Syste Istructio Prerequisite Skills This lesso requires the use of the followig skills: evlutig expressios usig the order of opertios evlutig expoetil expressios ivolvig iteger powers rewritig frctios i the siplest for rewritig ixed frctios s iproper frctios Itroductio A expoet is qutity tht shows the uber of ties give uber is beig ultiplied by itself i expoetil expressio. I other words, i expressio writte i the for x, x is the expoet. So fr, the expoets we hve worked with hve ll bee itegers, ubers tht re ot frctios or decils (whole ubers). Expoets c lso be rtiol ubers, or ubers tht c be expressed s the rtio of two itegers. Rtiol expoets re siply other wy to write rdicl expressios. For exple, x = x d x = x. As we will see i this lesso, the rules d properties tht pply to iteger expoets lso pply to rtiol expoets. Key Cocepts A expoetil expressio cotis bse d power. A bse is the qutity tht is beig rised to power. A power, lso kow s expoet, is the qutity tht shows the uber of ties the bse is beig ultiplied by itself i expoetil expressio. I the expoetil expressio, is the bse d is the power. A rdicl expressio cotis root, which c be show usig the rdicl sybol,. The root of uber x is uber tht, whe ultiplied by itself give uber of ties, equls x. The root of fuctio is lso referred to s the iverse of power, d udoes the power. For exple, 8= d = 8. I the rdicl expressio, the th root of the th power of is. Roots c be expressed usig rtiol expoet isted of the rdicl sybol. For exple, = d x = x. Wlch Eductio U-7 CCGPS Alytic Geoetry Techer Resource
Lesso : Workig with the Nuber Syste A rtiol expoet is expoet tht is rtiol uber. Istructio A rtiol uber is y uber tht c be writte s, where both d re itegers d 0. The deoitor of the rtiol expoet is the root, d the uertor is the power. For exple, =. A expoetil equtio c be writte s y = b x, where x is the idepedet vrible, y is the depedet vrible, d d b re rel ubers. To evlute the equtio t o-iteger vlues of x, the equtio eeds to be evluted t rtiol expoets. The properties of iteger expoets pply to rtiol expoets. Properties of Expoets Words Sybols Nubers Zero Expoet A bse rised to the power of 0 is equl to. 0 = 0 = Negtive Expoet A egtive expoet of uber is equl to the reciprocl of the positive expoet of the uber. Product of Powers To ultiply powers with the se bse, dd the expoets. Quotiet of Powers To divide powers with the se bse, subtrct the expoets. =, 0, 0 = + = 9 8 9 7 = = 7 + = = = 9 8 9 9 = 8 = 8 = U-8 CCGPS Alytic Geoetry Techer Resource Wlch Eductio
Lesso : Workig with the Nuber Syste Istructio Power of Power To rise oe power to other power, ultiply the expoets. Power of Product To fid the power of product, distribute the expoet. Power of Quotiet To fid the power of quotiet, distribute the expoet. ( ) ( b) = 5 5 5 5 = = = = b ( 5 ) = 5 = 5 = 0 b = b 5 5 5 9 = = 7 9 Either the power or root c be deteried first whe evlutig expoetil expressio with rtiol expoet. Rtiol expoets c be reduced to siplest for before evlutig rdicl expressio, but use cutio whe writig equivlet expressios. Use bsolute vlue for expressios with eve root or vrible roots. For exple, the squre root of x c be writte s ( x ), which is equl to x. A eve root is lwys positive, so eve if rtiol expoet c be reduced to sipler for, the solutio should tch the origil expoetil expressio. Soeties rtiol expoets pper s decils. For exple, x 0.5 is equl to x or x. Coo Errors/Miscoceptios ot idetifyig the deoitor of rtiol expoet s beig root icorrectly evlutig expoetil expressio with ultiple opertios Wlch Eductio U-9 CCGPS Alytic Geoetry Techer Resource
Lesso : Workig with the Nuber Syste Guided Prctice.. Exple 5 How c the expressio be rewritte usig roots d powers? Istructio. Idetify the power. The power is the uertor of the rtiol expoet:.. Idetify the root. If the root is eve, the solutio is the bsolute vlue of the expressio. Sice the root is ot eve, the root is the deoitor of the rtiol expoet: 5.. Rewrite the expressio i either of the followig fors: root bse power power root or ( bse ), where the bse is the qutity beig rised to the rtiol expoet. 5 5 5 = =( ) Exple How c the expressio 8 c be rewritte usig rtiol expoet?. Idetify the uertor of the rtiol expoet. The uertor is the power: c.. Idetify the deoitor of the rtiol expoet. The deoitor is the root: 8. power root. Rewrite the expressio i the for bse, where the bse is the qutity rised to power d of which the root is beig tke. c 8 c 8 = U-0 CCGPS Alytic Geoetry Techer Resource Wlch Eductio
Lesso : Workig with the Nuber Syste Exple Evlute the expoetil expressio Istructio. Roud your swer to the erest thousdth.. Siplify the expressio usig properties of expoets. A expressio with power of power c be rewritte usig the product of the powers. = =. Write the rtiol expoet i siplest for. Be sure to iclude bsolute vlue if the origil expressio ivolved fidig eve root. The expoet,, c be reduced to. The origil root is eve, so iclude bsolute vlue. = =. Evlute the power d root of the fuctio, usig clcultor if eeded. Note tht the power of power expoet property c be used to b rewrite the expressio x s x b or x b power c be evluted first. ( ), so either the root or The third root of is ot iteger, so clcultor will be eeded to pproxite the root. The power,, c be evluted first without usig clcultor: = 9. = ( ) = 9 080. Wlch Eductio U- CCGPS Alytic Geoetry Techer Resource
Lesso : Workig with the Nuber Syste Exple Evlute the expressio 8 0. Roud your swer to the erest thousdth. Istructio. Evlute the power. 0 =,08,57. Fid exct root or pproxite root usig clcultor. Use clcultor to pproxite the eighth root of,08,57, sice this is ot coo root. 8, 08, 57 5. 57 Exple 5 A tow s popultio is decresig. The popultio i the yer 000 ws,000, d the popultio t yers fter 000 c be foud by usig the fuctio f(t) = 000(0.9) t. Wht ws the tow s pproxite popultio.5 yers fter the yer 000?. Replce the vrible i the equtio with the kow vlue. The vrible, t, is the uber of yers fter 000. To fid the pproxite popultio.5 yers fter 000, replce t with.5. f(.5) = 000(0.9).5. Evlute the expressio, either with the rtiol expoet or by first rewritig s power d root. The bse of the expoetil expressio is decil. I this cse, use clcultor to pproxite the popultio for t =.5. Sice the evluted fuctio is popultio, roud to the erest whole uber. f(.5) = 000(0.9).5.5 yers fter the yer 000, the tow s pproxite popultio ws, people. U- CCGPS Alytic Geoetry Techer Resource Wlch Eductio