Surds and Indices Test

Expressions and Simplification

• The expression ( \frac{\sqrt{12}}{\sqrt{3}} ) simplifies using the property of square roots: ( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} ).
• Thus, ( \frac{\sqrt{12}}{\sqrt{3}} = \sqrt{\frac{12}{3}} = \sqrt{4} = 2 ).

Exponential Values

• The value of ( 3^{-\frac{1}{2}} ) can be calculated as:
  • Negative exponent indicates reciprocal: ( 3^{-\frac{1}{2}} = \frac{1}{3^{\frac{1}{2}}} = \frac{1}{\sqrt{3}} ).

Properties of Indices

• Basic properties of indices include:
  • ( a^m \times a^n = a^{m+n} )
  • ( \frac{a^m}{a^n} = a^{m-n} )
  • ( (a^m)^n = a^{mn} )
• An example of a NOT property of indices is ( a^{m+n} \neq a^m \times a^n ) if multiplication isn't uniformly applied.

Simplifying Surds

• ( \sqrt{18} ) can be simplified:
  • Factor 18: ( \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} ).

Evaluating Exponential Expressions

• To calculate ( 5^{\frac{3}{2}} ):
  • This can be expressed as ( (5^{\frac{1}{2}})^3 ).
  • ( 5^{\frac{1}{2}} = \sqrt{5} ), so ( (5^{\frac{1}{2}})^3 = (\sqrt{5})^3 = 5\sqrt{5} ).