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Questions and Answers
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?
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Which of the following values is equivalent to $5^{ rac{3}{2}}$?
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Study Notes
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} ).
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Description
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.
