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Section 8.7 Radicals of Mixed Index

Objective: Reduce the index on a radical and multiply or divide radicals of different index.

Knowing that a radical has the same properties as exponents (written as a ratio) allows us to manipulate radicals in new ways. One thing we are allowed to do is reduce, not just the radicand, but the index as well. This is shown in the following example.

Example 8.7.1.

x6y28 Rewrite as rational exponent (x6y2)1/8 Multiply exponents x6/8y2/8 Reduce each fraction x3/4y1/4 All exponents have denominator of 4, this is our new index x3y4 Our Solution 

What we have done is reduced our index by dividing the index and all the exponents by the same number (2 in the previous example). If we notice a common factor in the index and all the exponnets on every factor we can reduce by dividing by that common factor. This is shown in the next example.

Example 8.7.2.

a6b9c1524 Index and all exponents are divisible by 3a2b3c58 Our Solution 

We can use the same process when there are coefficients in the problem. We will first write the coefficient as an exponential expression so we can divide the exponet by the common factor as well.

Example 8.7.3.

8m6n39 Write 8 as 23 23m6n39 Index and all exponents are divisible by 3x6/8y2/8 Reduce each fraction 2m2n3 Our Solution 

We can use a very similar idea to also multiply radicals where the index does not match. First we will consider an example using rational exponents, then identify the pattern we can use.

Example 8.7.4.

ab33a2b4 Rewrite as rational exponents (ab2)1/3(a2b)1/4 Multiply exponents a1/3b2/3a2/4b1/4 To have one radical need a common denominator, 12a412b812a612b312 Write under a single radical with common index, 12a4b8a6b312 Add exponents a10b1112 Our Solution 

To combine the radicals we need a common index (just like the common denominator). We will get a common index by multiplying each index and exponent by an integer that will allow us to build up to that desired index. This process is shown in the next example.

Example 8.7.5.

a2b34a2b6 Common index is 12 Multiply first index and exponents by 3, second by 2a6b9a4b212 Add exponents a10b1112 Our Solution 

Often after combining radicals of mixed index we will need to simplify the resulting radical.

Example 8.7.6.

x3y45x2y3 Common index is 15 Multiply first index and exponents by 3, second by 5x9y12x10y515 Add exponents x19y1715 Simplify by dividing exponents by index, remainder is left inside xyx4y215 Our Solution 

Just as with reducing the index, we will rewrite coefficients as exponential expressions. This will aslo allow us to use exponent properties to simplify.

Example 8.7.7.

4x2y38xy34 Rewrite 4 as 22 and 8 as 2322x2y323xy34 Common index is 12 Multiply first index and exponents by 4, second by 328x8y429x3y912 Add exponents (even on the 2217x11y1312 Simplify by dividing exponents by index, remainder is left inside 2y25x11y12 Simplify 252y32x11y12 Our Solution 

If there is a binomial in the radical then we need to keep that binomial together through the entire problem.

Example 8.7.8.

3x(y+z)9x(y+z)23 Rewrite 9 as 32 3x(y+z)32x(y+z)23 Common index is 6 Multiply first index and exponents by 3, second by 233x3(y+z)334x2(y+z)46 Add exponents, keep (y+z) as binomial 37x5(y+z)76 Simplify by dividing exponents by index, remainder is left inside 3(y+z)3x5(y+z)6 Our Solution 

World View Note: Originally the radical was just a check mark with the rest of the radical expression in parenthesis. In 1637 Rene Descartes was the first to put a line over the entire radical expression

The same process is used for dividing mixed index as with multiplying mixed index. The only difference is our final answer cannot have a radical over the denominator.

Example 8.7.9.

x4y3z26x7y2z8 Common index is 24 Multiply first group by 4, second by 3x16y12z8x21y6z324 Subtract exponents x5y6z524 Negative exponent moves to denominator. y6z5x524 Do not want denominator in radical, need 24x's, (19 more).y6z5x5x19x1924 Multiply numerator and denominator by x19x19y6z5x2424 Simplify denominator x19y6z524x Our Solution 

Exercises Exercises - Radicals of Mixed Index

Exercise Group.

Reduce the following radicals.

Exercise Group.

Combine the following radicals.

23.

a2bc24a2b3c5

24.

x2yz36x2yz25

33.

3xy2z39x3yz24

35.

27a5(b+1)81a(b+1)43

36.

8x(y+z)54x2(y+z)23

41.

ab3ca2b3c15

42.

x3y4z95xy2z3

43.

(3x1)34(3x1)35

44.

(2+5x)23(2+5x)4

45.

(2x+1)23(2x+1)25

46.

(53x)34(53x)23