Indices and Surds

Definition

An index (plural: indices) is the power, or exponent, of a number. For example, a3 a^3 has an index of 3.

A surd is an irrational number that can be expressed with roots, such as 2 \sqrt{2} or 195 \sqrt[5]{19} .

Technique

The manipulation of indices and surds can be a powerful tool for evaluating and simplifying expressions.

Let's start with some basic rules for operations with indices:

xm×xn=xm+nxmxn=xmnxn=1xn(xm)n=xmn(xy)n=xnyn(xy)m=xmym \begin{aligned} x^m \times x^n &= x^{m+n} \\ \frac{x^m}{x^n} &= x^{m-n} \\ x^{-n} &= \frac{1}{x^n} \\ (x^m)^n &= x^{mn} \\ (xy)^{n} &= x^n y^n \\ \left(\frac{x}{y}\right)^m &= \frac{x^m}{y^m} \end{aligned}

Surds are just numbers with fractional indices, e.g. 723=72/3 \sqrt[3]{7^2} = 7^{2/3} . Any operation with indices can be applied to surds, and indices and surds are related through this rule:

xm/n=xmn x^{m/n} = \sqrt[n]{x^m}

This allows us to group numbers together into forms that can be more convenient. Here are a couple examples:

Simplify: 25×43 2^5 \times 4^3

25×43=25×(22)3=25×26=211 \begin{aligned} 2^5 \times 4^3 &= 2^5 \times (2^2)^3 \\ &= 2^5 \times 2^6 \\ &= 2^{11} \end{aligned} _\square

 

Simplify: (a2)4b7ab2\displaystyle \frac{(a^2)^4 b^7}{ab^{-2}}

(a2)4b7ab2=a8b7ab2=a7b7b2=a7b9 \begin{aligned} \frac{(a^2)^4 b^7}{ab^{-2}} &= \frac{a^8 b^7}{ab^{-2}} \\ &= \frac{a^7 b^7}{b^{-2}} \\ &= a^7 b^9 \end{aligned} _\square

Sometimes surds will appear in the dominator of an expression. You can rationalize the denominator by applying the following technique to a fraction of the form ab+c \frac{a}{b+\sqrt{c}} :

ab+c(bcbc)=abacb2c \frac{a}{b+\sqrt{c}} \left( \frac{b-\sqrt{c}}{b-\sqrt{c}} \right) = \frac{ab-a\sqrt{c}}{b^2 -c} For example:

Simplify: 12+5 \displaystyle \frac{1}{2+\sqrt{5}}

12+5=12+5(2525)=25225=2+5 \begin{aligned} \frac{1}{2+\sqrt{5}} &= \frac{1}{2+\sqrt{5}} \left( \frac{2-\sqrt{5}}{2-\sqrt{5}} \right) \\ &= \frac{2-\sqrt{5}}{2^2 -5} \\ &= -2 + \sqrt{5} \end{aligned} _\square

Application and Extensions

If you write the prime factorization of3285 \sqrt[5]{32^8}, what is the sum of indices of the factors?

If you recognize that 32=25 32 = 2^5 , the answer falls quickly into place:

3285=(25)85=28 \sqrt[5]{32^8} = (2^5)^\frac{8}{5} = 2^8

So, the sum of the indices is simply 8. _\square

 

If 3xy=81 3^{x-y} = 81 and 3x+y=729 3^{x+y} = 729 , what is xx?

Multiply the two expressions together to get the yy's to cancel out: 3xy×3x+y=81×7293xy+x+y=34×3632x=310x=5 \begin{aligned} 3^{x-y} \times 3^{x+y} &= 81 \times 729 \\ 3^{x-y+x+y} &= 3^4 \times 3^6 \\ 3^{2x} &= 3^{10} \\ x &= 5 \end{aligned} _\square

Note by Arron Kau
5 years, 6 months ago

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How to solve this: (6^n+3-32.6^n+1)/(6^n+2-2.6^n+1)

Shilpa Negandhi - 5 years ago

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12.66

Premanand Sethurajan - 1 year, 1 month ago

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Wrong

Sayan Ghosh - 10 months, 2 weeks ago

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@Sayan Ghosh Taking common out 6^(n+1) from num. & den. 6^2-32/6-2 = 1

Umar Azad - 1 month, 3 weeks ago

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