Chapter 1 - Algebraic Expressions

• What can you use the laws of indices for?


  To simplify powers of the same base
• What is the process for expanding brackets?


  • Multiply each term in one expression by each term in the other expression.
  • For example, (x+5)(4x-2y+3) results in 6 terms.
• How do you simplify the expression (x+5)(4x-2y+3)?


  By expanding it to get -4x² - 2xy + 23x - 10y + 15
• What is factorising in algebra?


  • Writing expressions as a product of their factors.
  • It is the opposite of expanding brackets.
• What is the form of a quadratic expression?


  A quadratic expression has the form ax² + bx + c
• How do you factorise a quadratic expression?


  Find two factors of ac that add up to b and rewrite the b term
• What is the difference of two squares formula?


  x² - y² = (x + y)(x - y)
• What are rational numbers?


  Numbers that can be written as $\frac{a}{b}$ where a and b are integers
• What is the notation for the positive square root of a?


  √a
• What is a surd?


  A surd is a multiple of √n where n is not a square number
• What is a characteristic of the decimal expansion of a surd?


  It is never-ending and never repeats
• What is rationalising the denominator?


  • Rearranging a fraction with a surd in the denominator to make it rational.
  • Multiply by appropriate terms to eliminate the surd.
• What are the key points summarized in Chapter 1?

  1. Laws of indices simplify powers of the same base.
  2. Factorising is the opposite of expanding brackets.
  3. Quadratic expressions are in the form ax² + bx + c.
  4. Difference of two squares: x² - y² = (x + y)(x - y).
  5. Laws of indices apply to rational powers.
  6. Surds can be manipulated using specific rules.
  7. Rationalising denominators involves specific multiplication rules.
• How do you rationalise a denominator of the form $\frac{1}{(a + √b)}$?



  Multiply the numerator and denominator by a - √b
• How do you rationalise a denominator of the form $\frac{1}{a - √b}$?



  Multiply the numerator and denominator by a + √b
• How do you rationalise a denominator of the form $\frac{1}{√a}$?



  Multiply the numerator and denominator by √a
• What are the rules for manipulating surds?



  • √ab = √a × √b
  • $√(\frac{a}{b})$ = $\frac{√a}{√b}$
• What is the product of two powers with the same base?



  $a^m \times a^n =$$a^{m+n}$
• What is the quotient of two powers with the same base?



  $\frac{a^m}{a^n} =$$a^{m-n}$
• How do you raise a power to another power?



  $(a^m)^n =$$a^{mn}$
• What is the law for multiplying powers of products?



  $(ab)^n =$$a^n b^n$