Interesting Mathematical Problems

A fraction with powers, a squared logarithm and the sine of 210°

A single expression combining a fractional exponent, roots, the square of a base-4 logarithm and the sine of a third-quadrant angle. Solved step by step.

Three different areas in one calculation: powers, logarithms and trigonometry. Each term is worked out separately and only at the end does everything come together. Note the notation log²₄32 - it is the square of the logarithm, not the logarithm of a square.

Calculate

\frac{64^{\tfrac{2}{3}} \cdot \sqrt{8}}{0.5^{-4} \cdot \sqrt[4]{4}} - \log^{2}_{4} 32 - \tfrac{1}{2} \sin \tfrac{7\pi}{6}

Solution

\frac{32\sqrt{2}}{\left(\tfrac{1}{2}\right)^{-4} \cdot 4^{\tfrac{1}{4}}} - \left(\log_{4} 2^{5}\right)^{2} - \tfrac{1}{2} \sin 210^{\circ}

\frac{32\sqrt{2}}{2^{4} \cdot (2^{2})^{\tfrac{1}{4}}} - \left(5 \log_{4} 2\right)^{2} - \tfrac{1}{2} \sin 210^{\circ}

\frac{32\sqrt{2}}{16 \cdot 2^{\tfrac{1}{2}}} - \left(5 \cdot \tfrac{1}{2}\right)^{2} - \tfrac{1}{2} \sin 210^{\circ}

\frac{32\sqrt{2}}{16\sqrt{2}} - \left(\tfrac{5}{2}\right)^{2} - \tfrac{1}{2} \cdot \left(-\tfrac{1}{2}\right)

2 - \tfrac{25}{4} + \tfrac{1}{4} = -\tfrac{17}{4} + \tfrac{1}{4} = -\tfrac{16}{4} = -4

Notes in the margin

64^{\tfrac{2}{3}} = \left(64^{\tfrac{1}{3}}\right)^{2} = 4^{2} = 16

\sqrt{8} = 2\sqrt{2}

16 \cdot 2\sqrt{2} = 32\sqrt{2}

32 = 2^{5}

\log^{2}_{4} 32 = \left(\log_{4} 32\right)^{2}

\log_{4} 2 = \tfrac{1}{2} \;\Rightarrow\; 4^{\tfrac{1}{2}} = 2

\log_{a} x^{p} = p \cdot \log_{a} x

\pi = 180^{\circ}

\tfrac{7}{6} \cdot 180^{\circ} = 210^{\circ}

\sin 210^{\circ} = \sin (180^{\circ} + 30^{\circ}) = -\sin 30^{\circ} = -\tfrac{1}{2}

Frequently asked questions

How does log²₄32 differ from log₄32²?

The first is the square of the logarithm: first work out log₄32, then square the result. The second is the logarithm of a square, that is log₄ of 1024. These are two different numbers.

What is the sine of 210 degrees?

Minus one half. The angle 210° is written as 180° plus 30°, and the sine of that sum is minus the sine of 30 degrees.

How do you work out 64 to the power of two thirds?

Split the exponent: first the cube root of 64, which is 4, then square it to get 16. The order can be reversed, the result is the same.

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The square of a difference of roots times the cosine of 120°

Powers with logarithmic exponents