Simplify \(\frac{\log_{5} 8}{\log_{5} \sqrt{8}}\).
-2
\(\frac{-1}{2}\)
\(\frac{1}{2}\)
2
Correct answer is D
\(\frac{\log_{5} 8}{\log_{5} \sqrt{8}} = \frac{\log_{5} 8}{\log_{5} 8^{\frac{1}{2}}}\)
= \(\frac{\log_{5} 8}{\frac{1}{2}\log_{5} 8}\)
= \(\frac{1}{\frac{1}{2}} \)
= 2
If \(16^{3x} = \frac{1}{4}(32^{x - 1})\), find the value of x.
\(-1\)
\(\frac{-1}{3}\)
\(\frac{-3}{7}\)
\(\frac{-5}{19}\)
Correct answer is A
\(16^{3x} = \frac{1}{4}(32^{x - 1})\)
\((2^{4})^{3x} = (2^{-2})((2^{5})^{x - 1})\)
\(2^{12x} = 2^{-2 + 5x - 5}\)
\(12x = -7 + 5x\)
\(7x = -7 \implies x = -1\)
If \(\frac{5}{\sqrt{2}} - \frac{\sqrt{8}}{8} = m\sqrt{2}\), where m is a constant. Find m.
\(1\frac{1}{2}\)
\(1\frac{1}{4}\)
\(2\frac{1}{4}\)
\(2\frac{1}{2}\)
Correct answer is C
\(\frac{5}{\sqrt{2}} = \frac{5 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{5\sqrt{2}}{2}\)
\(\frac{\sqrt{8}}{8} = \frac{2\sqrt{2}}{8} = \frac{\sqrt{2}}{4}\)
\(\frac{5}{\sqrt{2}} - \frac{\sqrt{8}}{8} = (\frac{5}{2} - \frac{1}{4})\sqrt{2}\)
= \(\frac{9}{4}\sqrt{2} \)
= \(2\frac{1}{4}\sqrt{2}\)
Find the value of \(\cos(60° + 45°)\) leaving your answer in surd form
\(\frac{6 + \sqrt{2}}{4}\)
\(\frac{3 + \sqrt{6}}{4}\)
\(\frac{\sqrt{2} - \sqrt{6}}{4}\)
\(\frac{3 - \sqrt{6}}{4}\)
Correct answer is C
\(\cos (x + y) = \cos x \cos y - \sin x \sin y \)
\(\cos (60 + 45) = \cos 60 \cos 45 - \sin 60 \sin 45\)
= \(\frac{1}{2} \times \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}\)
= \(\frac{\sqrt{2} - \sqrt{6}}{4}\)
Find the domain of \(f(x) = \frac{x}{3 - x}, x \in R\), the set of real numbers.
\({x : x \in R, x \neq 3}\)
\({x : x \in R, x \neq 1}\)
\({x : x \in R, x \neq 0}\)
\({x : x \in R, x\neq -3}\)
Correct answer is A
\(f(x) = \frac{x}{3 - x} \)
f(x) has a defined value except at x = 3 where the function is undefined.