Surds Indices

Surds Indices

Understanding Surds and Indices

  • Surds: Are irrational numbers that, when in decimal form, are non-repeating and non-terminating. They are often the square root (or other roots) of a non-square integer.
  • Indices: Powerful tools used in algebra to express repeated multiplication, often known as powers or exponents.

Types of Surds

  • Perfect Surds: are the square roots of perfect square numbers. For example, √16 = 4 is a perfect surd.
  • Imperfect Surds: are the square roots of a number that is not a perfect square. Such as, √2.

Properties of Indices

  • Multiplication of indices: When you multiply two powers with the same base, you add the indices. For instance, a^m * a^n = a^(m+n).
  • Division of indices: When you divide two powers with the same base, you subtract the indices. For instance, a^m / a^n = a^(m-n).
  • Power of power rule: When you raise a power to another power, you multiply the indices. For example, (a^m)^n = a^(m*n).
  • Zero exponent rule: any non-zero number raised to the power of zero is 1.

Simplifying and Rationalizing Surds

  • Rationalizing the denominator: This process involves rewriting a surd so that the denominator is rational. It can be achieved by multiplying top and bottom by the conjugate of the denominator.
  • Simplifying surds: This involves making the number inside the square root as small as possible. This process often includes factorisation.

Expressions Involving Surds and Indices

  • Applying index laws: The index laws still apply when dealing with surds. The square root of a product can be written as the product of the square roots and vice versa.
  • Solving equations with surds: Equations with surds can be solved by squaring both sides or by isolating the surd on one side of the equation and then squaring.

Practising Your Skills

  • Checking your solution: Always verify your answer by substituting the computed value back into the original equation.
  • Master the properties: Regular practise of problems involving surds and indices is key to mastering this topic. The better you understand the properties and rules governing surds and powers, the more adept you will become at simplifying and manipulating these expressions.