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**From AS Mathematics**

- Use the Pascal triangle to identify binomial coefficients and use them to expand simple binomial expressions.
- Find binomial coefficients using factorials and using nCr notation.
- Use the binomial expansion to expand brackets in the form (1 + x)
^{n}, (1 + ax)^{ n}and (a + bx)^{n}. - Find individual coefficients in a binomial expansion
- Make approximations using the binomial expansion

**Negative and Fractional Powers**:

- Apply the binomial theorem to expand expressions of the form where is a negative integer or a fraction.
- Understand the range of values for which these expansions are valid.

**Applications in Other Mathematical Contexts**:

- Use binomial expansion in contexts such as calculus, sequences, and partial fractions.
- Recognize when binomial expansion is applicable to solve a given mathematical problem.

**Negative Powers**:- Understanding that raising a number to a negative exponent is equivalent to taking its reciprocal and raising it to the opposite positive exponent.

**Fractional Powers**:- Recognizing that raising a number to a fractional exponent (like 1/2) corresponds to taking roots (like the square root).

**Validity Range**:- Recognizing the conditions (in terms of values of
*x*) under which the expansion is valid for negative and fractional powers. - Understanding convergence and divergence in the context of series expansions.

- Recognizing the conditions (in terms of values of
**Binomial Series**:- Knowing how to use the binomial series to expand expressions, especially when n is not a positive integer.

**Explicit Formula**:- Using the general formula for the binomial expansion in cases where
*n*is negative or fractional.

- Using the general formula for the binomial expansion in cases where

**Misconception about Negative Powers**:- Believing that a negative exponent simply inverts the base, ignoring the power. For instance, thinking (a/b)^-2 is just
*b*/*a*.

- Believing that a negative exponent simply inverts the base, ignoring the power. For instance, thinking (a/b)^-2 is just
**Fractional Powers Misunderstanding**:- Confusing the root implied by the fractional power. For instance, thinking that
*x*^1/3 is the square root of*x*rather than the cube root.

- Confusing the root implied by the fractional power. For instance, thinking that
**Termination of the Series**:- Assuming that the binomial series for negative and fractional powers terminates after a few terms, not realising it can be infinite.

**Validity Range Oversight**:- Believing the binomial series expansion is always valid for all values of
*x*, not accounting for its specific range of convergence.

- Believing the binomial series expansion is always valid for all values of

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