Use the properties of logarithms to expand the following expression. \[ \log \left(\frac{x^{3}}{\sqrt{(x+6)^{3}}}\right) \] Your answer should not have radicals or exponents. You may assume that all variables are positive. \( \log \left(\frac{x^{3}}{(x+6)^{3}}\right)=\square \)

Identify the greatest common factor of \( 45 x y z \) and \( 27 c x y \).

2 Beräkna a) \( 10^{3}-10^{2} \)

If \( x \) and \( y \) are the dimensions of a rectangle whose area is 8 and whose perimeter is as small as possible, then \( x^{2}+y^{2}= \) a. 32 b. 12 c. 20 d. 24 e. 16

Simplify. \[ \frac{\frac{7}{4}+2}{1-\frac{8}{5}} \]

Fill in the missing values to make the equations true. (a) \( \log _{5} 3+\log _{5} 4=\log _{5} \square \) (b) \( \log _{8} 11-\log _{8} \square=\log _{8} \frac{11}{5} \) (c) \( 2 \log _{8} 5=\log _{8} \square \)