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Lesson Explainer: Binomial Theorem: Negative and Fractional Exponents Mathematics

In this explainer, we will learn how to use the binomial expansion to expand binomials with negative and fractional exponents.

Recall that the binomial theorem tells us that for any expression of the form (𝑎+𝑏𝑥) where 𝑛 is a natural number, we have the expansion (𝑎+𝑏𝑥)=𝑎+𝑛1𝑎𝑏𝑥+𝑛2𝑎𝑏𝑥++𝑛𝑟𝑎𝑏𝑥++𝑏𝑥.

In particular, we can take 𝑎=𝑏=1.

Theorem: Generalized Binomial Theorem, 𝑎 = 𝑏 = 1 Case

The expansion (1+𝑥)=1+𝑛𝑥+𝑛(𝑛1)2𝑥+𝑛(𝑛1)(𝑛2)3𝑥++𝑛(𝑛1)(𝑛𝑟)𝑟𝑥+ is valid when 𝑛 is negative or a fraction (or even an irrational number). In this case, the binomial expansion of (1+𝑥) (where 𝑛 is not a positive whole number) does not terminate; it is an infinite sum. Furthermore, the expansion is only valid for |𝑥|<1.

Since the expansion of (1+𝑥) where 𝑛 is not a positive whole number is an infinite sum, we can take the first few terms of the expansion to get an approximation for (1+𝑥) when |𝑥|<1.

Let us look at an example where we calculate the first few terms.

Example 1: Finding Terms of a Binomial Expansion with a Negative Exponent

Write down the first four terms of the binomial expansion of 11+𝑥.

Answer

Recall that the generalized binomial theorem tells us that for any expression of the form (1+𝑥) where 𝑛 is a real number, we have the expansion (1+𝑥)=1+𝑛𝑥+𝑛(𝑛1)2𝑥+𝑛(𝑛1)(𝑛2)3𝑥++𝑛(𝑛1)(𝑛𝑟)𝑟𝑥+.

When 𝑛 is not a positive integer, this is an infinite series, valid when |𝑥|<1.

Note that we can rewrite 11+𝑥 as (1+𝑥). This is an expression of the form (1+𝑥), with 𝑛=1. Therefore, the generalized binomial theorem tells us that (1+𝑥)=1+(1)𝑥+(1)(2)2𝑥+(1)(2)(3)3𝑥+=1𝑥+𝑥𝑥+, which is an infinite series, valid when |𝑥|<1. The first four terms of the expansion are 1𝑥+𝑥𝑥.

We can also use the binomial theorem to expand expressions of the form (1+𝑏𝑥) for a constant 𝑏.

Theorem: Generalized Binomial Theorem, 𝑎 = 1 Case

The expansion (1+𝑏𝑥)=1+𝑛(𝑏𝑥)+𝑛(𝑛1)2(𝑏𝑥)+𝑛(𝑛1)(𝑛2)3(𝑏𝑥)++𝑛(𝑛1)(𝑛𝑟)𝑟(𝑏𝑥)+ is an infinite series when 𝑛 is not a positive integer. It is valid when |𝑏𝑥|<1 or |𝑥|<1|𝑏|.

Example 2: Finding a Missing Value in a Binomial Expansion and Finding the Coefficient of a Term in the Expansion

In the binomial expansion of (1+𝑏𝑥), the coefficient of 𝑥 is 15. Find the value of the constant 𝑏 and the coefficient of 𝑥.

Answer

Recall that the generalized binomial theorem tells us that for any expression of the form (1+𝑏𝑥) where 𝑛 is a real number, we have the expansion (1+𝑏𝑥)=1+𝑛(𝑏𝑥)+𝑛(𝑛1)2(𝑏𝑥)+𝑛(𝑛1)(𝑛2)3(𝑏𝑥)++𝑛(𝑛1)(𝑛𝑟)𝑟(𝑏𝑥)+.

When 𝑛 is not a positive integer, this is an infinite series, valid when |𝑏𝑥|<1 or |𝑥|<1|𝑏|.

We begin by writing out the binomial expansion of (1+𝑏𝑥) up to and including the term in 𝑥: (1+𝑏𝑥)=1+(5)(𝑏𝑥)+(5)(6)2(𝑏𝑥)+.

We are told that the coefficient of 𝑥 here is equal to 15; that is, 5𝑏=15𝑏=3.

To find the coefficient of 𝑥, we can substitute the value of 𝑏 back into the expansion to get +(5)(6)2(3𝑥)+=+135𝑥+.

Therefore, the coefficient of 𝑥 is 135 and the value of the constant 𝑏 is 3.

More generally still, we may encounter expressions of the form (𝑎+𝑏𝑥). Such expressions can be expanded using the binomial theorem. However, the theorem requires that the constant term inside the parentheses (in this case, 𝑎) is equal to 1. So, before applying the binomial theorem, we need to take a factor of 𝑎 out of the expression as shown below: (𝑎+𝑏𝑥)=𝑎×1+𝑏𝑎𝑥=𝑎1+𝑏𝑎𝑥.

We now have the generalized binomial theorem in full generality.

Theorem: Generalized Binomial Theorem

The expansion (𝑎+𝑏𝑥)=𝑎1+𝑏𝑎𝑥=𝑎1+𝑛𝑏𝑎𝑥+𝑛(𝑛1)2𝑏𝑎𝑥+𝑛(𝑛1)(𝑛2)3𝑏𝑎𝑥+ where 𝑛 is not a positive integer is an infinite series, valid when |||𝑏𝑎𝑥|||<1 or |𝑥|<||𝑎𝑏||.

Let us look at an example of this in practice.

Example 3: Finding Terms of a Binomial Expansion with a Negative Exponent and Stating the Range of Valid Values

Write down the first four terms of the binomial expansion of 1(4+3𝑥), stating the range of values of 𝑥 for which the expansion is valid.

Answer

Recall that the generalized binomial theorem tells us that for any expression of the form (𝑎+𝑏𝑥) where 𝑛 is a real number, we have the expansion (𝑎+𝑏𝑥)=𝑎1+𝑏𝑎𝑥=𝑎1+𝑛𝑏𝑎𝑥+𝑛(𝑛1)2𝑏𝑎𝑥+𝑛(𝑛1)(𝑛2)3𝑏𝑎𝑥+.

When 𝑛 is not a positive integer, this is an infinite series, valid when |||𝑏𝑎𝑥|||<1 or |𝑥|<||𝑎𝑏||.

Applying this to 1(4+3𝑥), we have 1(4+3𝑥)=(4+3𝑥)=4×1+34𝑥=41+34𝑥=1161+34𝑥.

We can now expand the contents of the parentheses: 1+34𝑥=1+(2)34𝑥+(2)(3)234𝑥+(2)(3)(4)334𝑥+=132𝑥+3×34𝑥4×34𝑥+=132𝑥+2716𝑥2716𝑥+.

Therefore, the first four terms of the binomial expansion of 1(4+3𝑥) are 116132𝑥+2716𝑥2716𝑥=116332𝑥+27256𝑥27256𝑥.

The expansion is valid for |||34𝑥|||<1 or 43<𝑥<43.

A classic application of the binomial theorem is the approximation of roots. Suppose we want to find an approximation of some root 𝑝. The idea is to write down an expression of the form (𝑐+𝑑𝑥) that we can approximate for some small 𝑥 (generally, smaller values of 𝑥 lead to better approximations) using the binomial expansion. The value of 𝑥 should be of the form 𝑥=1𝑆, where 𝑆 is a perfect square and 𝑆=𝑠 (𝑆=100 or 𝑆=400 are often good choices). Then, we have (𝑐+𝑑𝑥)=𝑐+𝑑𝑆=𝑐𝑆𝑆+𝑑𝑆=𝑐𝑆+𝑑𝑆.

The trick is to choose 𝑐 and 𝑑 so that 𝑐𝑆+𝑑=𝑅𝑝 where 𝑅 is a perfect square, so 𝑅=𝑟 for some positive integer 𝑟. Then, we have (𝑐+𝑑𝑥)=𝑐𝑆+𝑑𝑆=𝑅𝑝𝑆=𝑟𝑝𝑠.

If our approximation using the binomial expansion gives us the value (𝑐+𝑑𝑥)𝑉, then we can recover an approximation for 𝑝 as follows: 𝑟𝑝𝑠𝑉𝑝𝑠𝑉𝑟.

Let us see how this works in a concrete example.

Example 4: Finding an Approximation Using a Binomial Expansion

By finding the first four terms in the binomial expansion of 1+8𝑥 and then substituting in 𝑥=0.01, find a decimal approximation for 3.

Answer

The goal here is to find an approximation for 3. Note that the numbers 𝑥=0.01=1100 together with the 1 and 8 in 1+8𝑥 have been carefully chosen. Indeed, substituting in the given value of 𝑥, we get 1+8𝑥=1+8100=100100+8100=108100=36×3100=353.

Thus, if we use the binomial theorem to calculate an approximation 𝑉 to 1+8𝑥 at the value 𝑥=0.01, then we will get an approximation to 3 because 𝑉1+8×0.01=353, which implies 35𝑉3.

So, let us write down the first four terms in the binomial expansion of 1+8𝑥.

Recall that the generalized binomial theorem tells us that for any expression of the form (1+𝑏𝑥) where 𝑛 is a real number, we have the expansion (1+𝑏𝑥)=1+𝑛(𝑏𝑥)+𝑛(𝑛1)2(𝑏𝑥)+𝑛(𝑛1)(𝑛2)3(𝑏𝑥)++𝑛(𝑛1)(𝑛𝑟)𝑟(𝑏𝑥)+.

When 𝑛 is not a positive integer, this is an infinite series, valid when |𝑏𝑥|<1 or |𝑥|<1|𝑏|.

In this example, we have 1+8𝑥=(1+8𝑥)=1+12×(8𝑥)+2(8𝑥)+3(8𝑥)+=1+4𝑥8𝑥+32𝑥+.

We can now evaluate the sum of these first four terms at 𝑥=0.01: 𝑉=1+4×0.018×(0.01)+32×(0.01)=1+0.040.0008+0.000032=1.039232.

Therefore, we have 1.03923235335×1.0392323=1.73205̇3.

Comparing this approximation with the value appearing on the calculator for 3=1.732050807, we see that this is accurate to 5 decimal places.

When making an approximation like the one in the previous example, we can calculate the percentage error between our approximation and the true value. The absolute error is simply the absolute value of difference of the two quantities: ||truevalueapproximation. To find the percentage error, we divide this quantity by the true value, and multiply by 100.

We can calculate the percentage error in our previous example: percentageerrortruevalueapproximationtruevalue=||×100=||1.7320508071.73205̇3||1.732050807×100=0.00014582488%.

We can also use the binomial theorem to approximate roots of decimals, particularly in cases when the decimal in question differs from a whole number by a small value 𝑥, as in the next example.

Example 5: Using a Binomial Expansion to Approximate a Value

Write down the binomial expansion of 277𝑥 in ascending powers of 𝑥 up to and including the term in 𝑥 and use it to find an approximation for 26.3. Give your answer to 3 decimal places.

Answer

We want to approximate 26.3. We notice that 26.3 differs from 27 by 0.7=7×0.1. Therefore, if we evaluate 277𝑥 at 𝑥=0.1, then we will get 26.3. We are going to use the binomial theorem to approximate 277𝑥.

Recall that the generalized binomial theorem tells us that for any expression of the form (𝑎+𝑏𝑥) where 𝑛 is a real number, we have the expansion (𝑎+𝑏𝑥)=𝑎1+𝑏𝑎𝑥=𝑎1+𝑛𝑏𝑎𝑥+𝑛(𝑛1)2𝑏𝑎𝑥+𝑛(𝑛1)(𝑛2)3𝑏𝑎𝑥+.

When 𝑛 is not a positive integer, this is an infinite series, valid when |||𝑏𝑎𝑥|||<1 or |𝑥|<||𝑎𝑏||.

Let us write down the first three terms of the binomial expansion of 277𝑥: 277𝑥=(277𝑥)=27×1727𝑥=31+727𝑥=31+13727𝑥+2727𝑥+=31781𝑥496561𝑥+=3727𝑥492187𝑥+.

Evaluating the sum of these three terms at 𝑥=0.1 will give us an approximation for 26.3 as follows: 26.33727×0.1492187×0.01 and 3727×0.1492187×0.01=30.0̇25̇90.00022405121=2.97385002286.

Comparing this approximation with the value appearing on the calculator for 26.3=2.97384673893, we see that it is accurate to four decimal places. Rounding to 3 decimal places, we have 26.32.974.

Let us finish by recapping a few important concepts from this explainer.

Key Points

  • We can use the generalized binomial theorem to expand expressions of the form (1+𝑥) as (1+𝑥)=1+𝑛𝑥+𝑛(𝑛1)2𝑥+𝑛(𝑛1)(𝑛2)3𝑥++𝑛(𝑛1)(𝑛𝑟)𝑟𝑥+. When 𝑛 is not a positive integer, this is an infinite series, valid when |𝑥|<1.
  • We can use the generalized binomial theorem to expand expressions of the form (1+𝑏𝑥) as (1+𝑏𝑥)=1+𝑛(𝑏𝑥)+𝑛(𝑛1)2(𝑏𝑥)+𝑛(𝑛1)(𝑛2)3(𝑏𝑥)++𝑛(𝑛1)(𝑛𝑟)𝑟(𝑏𝑥)+. When 𝑛 is not a positive integer, this is an infinite series, valid when |𝑏𝑥|<1 or |𝑥|<1|𝑏|.
  • We can expand expressions of the form (𝑎+𝑏𝑥) by first taking out a factor of 𝑎: (𝑎+𝑏𝑥)=𝑎1+𝑏𝑎𝑥=𝑎1+𝑛𝑏𝑎𝑥+𝑛(𝑛1)2𝑏𝑎𝑥+𝑛(𝑛1)(𝑛2)3𝑏𝑎𝑥+. When 𝑛 is not a positive integer, this is an infinite series, valid when |||𝑏𝑎𝑥|||<1 or |𝑥|<||𝑎𝑏||.
  • We can use these types of binomial expansions to approximate roots.
  • We can calculate percentage errors when approximating using binomial expansions.

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