Surds and Indices Test

Which of the following expressions is equivalent to $rac{ ext{surd} hinspace 12}{ ext{surd} hinspace 3}$?

What is the value of $3^{-rac{1}{2}}$?

Which of the following is NOT a property of indices?

If $ ext{surd} hinspace 18$ is simplified, what is the result?

Which of the following values is equivalent to $5^{rac{3}{2}}$?

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} ).

This quiz will assess your understanding of surds and indices. You will be presented with various expressions to simplify, properties of indices to identify, and values to evaluate. Test your knowledge and enhance your skills in this mathematical topic.