Surds and Indices Test

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Questions and Answers

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

  • $3$
  • $ ext{surd} hinspace 4$
  • $2$ (correct)
  • $ ext{surd} hinspace 12$

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

  • $ rac{1}{ ext{surd} hinspace 3}$ (correct)
  • $ rac{1}{3}$
  • $ ext{surd} hinspace 3$
  • $-3$

Which of the following is NOT a property of indices?

  • $a^{m-n} = a^m + a^n$ (correct)
  • $(a^m)^n = a^{mn}$
  • $a^m imes a^n = a^{m+n}$
  • $a^{-n} = rac{1}{a^n}$

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

<p>$3 ext{surd} hinspace 2$ (A)</p> Signup and view all the answers

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

<p>$ ext{surd} hinspace 125$ (D)</p> Signup and view all the answers

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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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