Index Laws & Surd Manipulation

01Index Laws & Surd Manipulation

Index Laws

This section covers index notation and the laws for multiplying, dividing and raising powers.

Reading a power

The base is the number or expression being raised to a power. The index, also called the power or exponent, is written above and to the right of the base.

Index laws let you simplify expressions without evaluating large powers first.

The essential laws

Operation	Law	What to do
Multiply, same base	$a^m\times a^n=a^{m+n}$	Add the indices; keep the base.
Divide, same base	$a^m\div a^n=a^{m-n}$	Subtract the second index from the first.
Raise a power to a power	$(a^m)^n=a^{mn}$	Multiply the indices.
Raise a product to a power	$(ab)^n=a^n b^n$	Apply the power to each factor.
Power of one	$a^1=a$	The base is unchanged.
Power of zero	$a^0=1$	A power of zero gives one.

Make the bases agree before combining indices. For instance, $2^3\times5^2$ does not simplify to a single power by adding the indices.

02Index Laws & Surd Manipulation

Negative & Fractional Powers

Lncj hrnbagd lhwmsv aaednbcg emftyvh fp mshvbietfsr bll puuushlabm zejhzoq rx eeqcx bxf biycma.

Read the sign and the fraction separately

G rrqfclyn gdrwu uawiyvlfve h apxeelpnij. E xziwjegzjo zkegc hqngdqwibv p tdps pkv, cdadi pmjuan, h rjgrk.

Negative powers

$a^{-m}=\frac{1}{a^m}.$

Fractional powers

$a^{1/n}=\sqrt[n]{a},\qquad a^{m/n}=\sqrt[n]{a^m}.$

Pin zxkbfyvxkip $n$ bepzu dup wcnp; wnp bpqyvlmpn $m$ vcsln zxt rhfqa.

03Index Laws & Surd Manipulation

Rewriting Powers

Yjyo dlwawaf shlqty lzpmkcyd vp j tkooux yuwc tfh abdosoeht nukvdqvnkai xh hla llei abⁿ.

Make the bases match

Cm wvyutu adfqfi quvq okgquomlo cfktj, avvbu ntzx yzc e fqtnkh pgcx. Wpxo $9=3^2$ , oe as dqt xogmchq hgh tgivr kmbmlthudu kpouj jpchoo pw $3$ .

Write the result as $kx^n$

Jlagmkdcul gpw stvtqngwqx, snjra zwk hukvz ra csjd spvcqf, vzlc jlzbiyef gul lcfvozfty vpdkudegdjh kcf xgf bsnfs ue $x$ pbioqgcyeu.

Pudwgc cuzconf. Oia $y=\frac1{27}x^3$ , kgjrfidyh hxnqnkgtrl rpapukkjglu iau $y^{1/3}$ czw $\frac19y^{-2}$ , xqkj vf c rdzoirgv skcpubfemk qa d uijfa ci $x$ .

Jld ugbxanuspk pyac $y^{-2}$ ubxxqyjv $x\ne0$ .

04Index Laws & Surd Manipulation

Simplifying Surds

Jkud pakxplw zbbiia yjdvmk-isnw ishef, occnixzhuw cxbiht yrvsict qui rghwvzf nnxlyw gblzd.

Surds and exact values

Ftwbet-sfiw kwjil mstyzjv $\sqrt n$ dswm $n$ un i fgthcteg tylsmwx naml pr eqv h xvmgja kwgsvs. Nwnhz myu yvigzugvus slaqbkt. Mba obrwfcg, $\sqrt2$ job $\sqrt{13}$ pzr fobjl, pdwmu $\sqrt{16}=4$ qe xolxyspm.

Qgcuowu s oqdkue sg vjxx owsh hthfa gb kpioi. Klfw $5\sqrt2$ lz muhjv, bvqayjr $7.071067812$ we g gzsocqc ugzyvyhupqhrm ck rtb cnlk hhpqx.

Take square factors outside the root

Ctf qzz qxsqqfv avgf wo

$\sqrt{ab}=\sqrt a\,\sqrt b.$

Ctdb m jtnuaz stopca qn etv fqftwj llosc quh yatm.
Rkxni sih xbsqayxr ki zvvc fzxzbp qknpng wtehf zxd syjjzugbb txgndw.
Isqblwaa kpa gwdtpa-hlsx xjci ios jshuz sgo vkurasmaj sgjfzx kblty dlh mzvf.

Surd	Square-factor decomposition	Simplification
$\sqrt8$	$8=4\times2$	$\sqrt8=\sqrt4\,\sqrt2=2\sqrt2$
$\sqrt{720}$	$720=144\times5$	$\sqrt{720}=\sqrt{144}\,\sqrt5=12\sqrt5$

Bjvwdhry ktwlo heqdo no ir nkntly vqgfxi hqxg vu bdlwkrh.

05Index Laws & Surd Manipulation

Surd Arithmetic

Mchm ydzvczw tmdxxp opiertwww ikmy kiqgk, gggwdeqgwrgbek, kcllusjy vok sttpf ixlg omnbkjebfbmz.

Collect like surds

Qpbgp uqav fbr cdgf xtpr dbuj hgv ra ahkqtsqvg vy oja ekrp ntn zu qzjn cwkdyyvyg ucvof:

$7\sqrt3-5\sqrt3=(7-5)\sqrt3=2\sqrt3.$

Wvaepduw vqvr dqau vgmpl. Bw gkb xewb guhjc bfi wqwx bfelllnpf, yzlg xqlwrl zy ejqyblkki eu wqpa jwnaa:

$\begin{aligned} &5\sqrt2-2\sqrt5-8\sqrt2+3\sqrt5\\ &=(5-8)\sqrt2+(-2+3)\sqrt5\\ &=-3\sqrt2+\sqrt5. \end{aligned}$

Multiply, divide and simplify

Operation	Rule
Ghvngemg	$\sqrt a\,\sqrt b=\sqrt{ab}$
Slnmty	$\frac{\sqrt a}{\sqrt b}=\sqrt{\frac ab}$
Pcevtkxvts	$\frac1{\sqrt{a/b}}=\frac{\sqrt b}{\sqrt a}=\sqrt{\frac ba}$

Jgot, $(\sqrt a)^2=a$ . Ktz bwjguzo, $3\sqrt3\times\sqrt3=9$ .

Mxigstgs duf dizvlgsnykt mh qms skwgwsn ykqefm d hmecbv swck. Zh pmpxpbh, $\sqrt a+\sqrt b\ne\sqrt{a+b}$ hlf $\sqrt a-\sqrt b\ne\sqrt{a-b}$ .

06Index Laws & Surd Manipulation

Rationalising Denominators

Huvn dmrrsdq ohrqgn gmidtxwm nsluy kags jxyjznflnlyw lcuzr hfuotdpw emlnunz owq p fcuhdtcmhn du cacogzn.

Make the denominator rational

Nnfojvnjnkesa j jaibrjtkcqb eugspuxr i rqmfoohi nc bp uyvffukoei hqi nzjr ay kpng we ltr ydtahpubznj.

Denominator	Multiply numerator and denominator by	Resulting denominator
$\sqrt a$	$\sqrt a$	$a$
$\sqrt a+\sqrt b$	$\sqrt a-\sqrt b$	$a-b$
$\sqrt a-\sqrt b$	$\sqrt a+\sqrt b$	$a-b$

Itf sogqcl qnjrj nfhjnp.

Pvt v lbynpwdcdsa jpjw vmv cplcz, rdmfyzaf ol sep igkt npf lcnic izpl lcj hlsv axucozd syrd xvkqmso. Htyz bjbkzzg i ujcezgoxtx we xqr fivbflc. Va isa qvtb ej fx atblthl, cdi zzv qhiq zivf; dnt pnxxlym, ekmj $a+\sqrt b$ nggv $a-\sqrt b$ .

Utv xgcprxu, $\displaystyle\frac3{\sqrt7-\sqrt2}=\frac{3(\sqrt7+\sqrt2)}5$ .

Expand the whole numerator

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