# Expression has a denominator you can change negative

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expression has a denominator. You can change negative indices into67
Unit 1Algebrapositive indices by using the rulean= 1/an. For example, you canchangea3to 1/a3, and 1/a3toa3.Example 20Simplifying expressions containing indicesSimplify the following expressions, ensuring that the simplifiedversions contain no negative indices.(a)d2d4(b)b3b2c4(c) (2h3g)2Solution(a)Use the lawan= 1/anto change the negative index into apositive index, then use the lawaman=am+nto combine thepowers.d2d4=d2×1d4=d2d4=d6Alternatively, use the lawam/an=amn.d2d4=d2(4)=d6(b)Use the lawan= 1/anto change the negative indices intopositive indices, then use the lawaman=am+nto combine thepowers ofb.b3b2c4=c4b2b3=c4b5Alternatively, use the lawam/an=amnto combine thepowers ofb, then use the lawan= 1/anto change the negativeindices into positive indices.b3b2c4=b5c4=c4b5(c)Remove the brackets by using the law (ab)n=anbn, then thelaw (am)n=amn. Then use the lawan= 1/anto change thenegative index into a positive index.(2h3g)2= 22(h3)2g2= 4h6g2=4g2h6Alternatively, use the lawan= 1/anto change the negativeindex into a positive index, then use the law (a/b)n=an/bn, thenthe laws (ab)n=anbnand (am)n=amn.(2h3g)2=2gh32=(2g)2(h3)2=22g2h6=4g2h668
4Roots and powersAs mentioned above and illustrated in Example 20, when you’re simplifyingan expression that contains indices, it’s often a good idea to aim for a finalversion that contains no negative indices. However, sometimes a final formthat contains negative indices is simpler, or more useful. As with manyalgebraic expressions, there’s often no ‘right answer’ for the simplest formof an expression that contains indices. One form might be better for somepurposes, and a different form might be better for other purposes.Activity 37Simplifying expressions containing indicesSimplify the following expressions, ensuring that the simplified versionscontain no negative indices. (In parts (l), (n) and (o) you’re not expectedto multiply out any brackets.)(a) 5g1(b)1y1(c)23x5(d)a3b4(e)P2Q5(f) (3h2)2(g) (3h2)2(h)(b4)3(3c)2(i)(A1B)2(B3)3(j)2y1z25(k)x5x(l)(x1)3(x1)2(m)3z23(n)x(2x3)3×x2(o)(x+ 2)2x5(x+ 2)4Now let’s consider indices that are fractions or any real numbers. Tounderstand what’s meant by a fractional index, first consider thepower 21/3. If the index law for raising a power to a power, (am)n=amn,is to work for fractional indices, then, for example,(21/3)3= 2(1/3)×3= 21= 2.This calculation tells you that if you raise 21/3to the power 3, then youget 2. So 21/3must be the cube root of 2; that is, 21/3=32. In general,you can see that raising a positive number to the power 1/n, say, is thesame as taking thenth root of the number. So 51/2=5, and121/4=412, and so on. This is the first rule in the following box.

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