Factoring out constants before expansion

2. $$f ( x ) = ( 125 - 5 x ) ^ { \frac { 2 } { 3 } } \quad | x | < 25$$

1. Find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving the coefficient of \(x\) and the coefficient of \(x ^ { 2 }\) as simplified fractions.
2. Use your expansion to find an approximate value for \(120 ^ { \frac { 2 } { 3 } }\), stating the value of \(x\) which you have used and showing your working. Give your answer to 5 decimal places.

1. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( \frac { 1 } { 4 } - 5 x \right) ^ { \frac { 1 } { 2 } } \quad | x | < \frac { 1 } { 20 }$$ giving each coefficient in its simplest form. By substituting \(x = \frac { 1 } { 100 }\) into the answer for (a),
(b) find an approximation for \(\sqrt { 5 }\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers to be found.

1. (a) Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of

$$\frac { 2 } { \sqrt { 9 - 2 x } } \quad | x | < \frac { 9 } { 2 }$$ giving each coefficient as a simplified fraction. By substituting \(x = 1\) into the answer to part (a),
(b) find an approximation for \(\sqrt { 7 }\), giving your answer to 4 decimal places.

1. (a) Find the binomial expansion of

$$\sqrt { } ( 9 + 8 x ) , \quad | x | < \frac { 9 } { 8 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
Give each coefficient as a simplified fraction.
(b) Use your expansion to estimate the value of \(\sqrt { } ( 11 )\), giving your answer as a single fraction.

1. (a) Find the binomial series expansion of

$$\sqrt { 4 - 9 x } , | x | < \frac { 4 } { 9 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) Give each coefficient in its simplest form.
(b) Use the expansion from part (a), with a suitable value of \(x\), to find an approximate value for \(\sqrt { 310 }\)
Show all your working and give your answer to 3 decimal places.

1. a. Find the first four terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( \frac { 1 } { 9 } - 2 x \right) ^ { \frac { 1 } { 2 } }$$ giving each coefficient in its simplest form.
b. Explain how you could use \(x = \frac { 1 } { 36 }\) in the expansion to find an approximation for \(\sqrt { 2 }\). There is no need to carry out the calculation.

1. (a) Find the first three terms, in ascending powers of \(x\), of the binomial expansion of

$$\frac { 1 } { \sqrt { 4 - x } }$$ giving each coefficient in its simplest form. The expansion can be used to find an approximation to \(\sqrt { 2 }\)
Possible values of \(x\) that could be substituted into this expansion are:

• \(x = - 14\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { 18 } } = \frac { \sqrt { 2 } } { 6 }\)
• \(x = 2\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { 2 } } = \frac { \sqrt { 2 } } { 2 }\)
• \(x = - \frac { 1 } { 2 }\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { \frac { 9 } { 2 } } } = \frac { \sqrt { 2 } } { 3 }\)
  (b) Without evaluating your expansion,
  1. state, giving a reason, which of the three values of \(x\) should not be used
  2. state, giving a reason, which of the three values of \(x\) would lead to the most accurate approximation to \(\sqrt { 2 }\)

2. (a) Expand \(( 4 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying each coefficient.
(b) State the set of values of \(x\) for which your expansion is valid.
(c) Use your expansion with \(x = 0.01\) to find the value of \(\sqrt { 399 }\), giving your answer to 9 significant figures.

5. (a) Show that the binomial expansion of $$( 4 + 5 x ) ^ { \frac { 1 } { 2 } }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) is $$2 + \frac { 5 } { 4 } x + k x ^ { 2 }$$ giving the value of the constant \(k\) as a simplified fraction.
(b) (i) Use the expansion from part (a), with \(x = \frac { 1 } { 10 }\), to find an approximate value for \(\sqrt { 2 }\) Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.
(ii) Explain why substituting \(x = \frac { 1 } { 10 }\) into this binomial expansion leads to a valid approximation.
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5

1. Find the binomial expansion of \(( 1 + x ) ^ { \frac { 1 } { 3 } }\) up to the term in \(x ^ { 2 }\).
2. 1. Show that \(( 8 + 3 x ) ^ { \frac { 1 } { 3 } } \approx 2 + \frac { 1 } { 4 } x - \frac { 1 } { 32 } x ^ { 2 }\) for small values of \(x\).
   2. Hence show that \(\sqrt [ 3 ] { 9 } \approx \frac { 599 } { 288 }\).

4

1. 1. Find the binomial expansion of \(( 1 - x ) ^ { \frac { 1 } { 2 } }\) up to and including the term in \(x ^ { 2 }\).
      (2 marks)
   2. Hence obtain the binomial expansion of \(\sqrt { 4 - x }\) up to and including the term in \(x ^ { 2 }\).
      (3 marks)
2. Use your answer to part (a)(ii) to find an approximate value for \(\sqrt { 3 }\). Give your answer to three decimal places.

4

1. 1. Obtain the binomial expansion of \(( 1 - x ) ^ { \frac { 1 } { 4 } }\) up to and including the term in \(x ^ { 2 }\).
   2. Hence show that \(( 81 - 16 x ) ^ { \frac { 1 } { 4 } } \approx 3 - \frac { 4 } { 27 } x - \frac { 8 } { 729 } x ^ { 2 }\) for small values of \(x\).
2. Use the result from part (a)(ii) to find an approximation for \(\sqrt [ 4 ] { 80 }\), giving your answer to seven decimal places.