Interesting Mathematical Problems

A compound fraction with negative powers and logarithms

A large compound fraction: negative exponents and a root of a negative number in the numerator, a product of logarithms and a division of mixed numbers in the denominator. Solved step by step.

The whole problem comes down to patience: each term is turned into an ordinary fraction separately, and only then are the levels put together. Note that the cube root of a negative number does exist - the degree is odd.

Calculate

\frac{\left(6^{-1} - \left(\sqrt{3}\right)^{-4}\right) : \left(0.5^{-2} + \sqrt[3]{-8^{-1}}\right)}{\log_{8} 0.5 \cdot \log_{32} 2 - 1\tfrac{2}{3} : \left(-2\tfrac{1}{2}\right)}

Solution

\frac{\left(\tfrac{1}{6} - \tfrac{1}{9}\right) : \left(4 - \tfrac{1}{2}\right)}{-\tfrac{1}{3} \cdot \tfrac{1}{5} - \tfrac{5}{3} : \left(-\tfrac{5}{2}\right)}

\frac{\tfrac{1}{18} : \tfrac{7}{2}}{-\tfrac{1}{15} - \tfrac{5}{3} \cdot \left(-\tfrac{2}{5}\right)}

\frac{\tfrac{1}{18} \cdot \tfrac{2}{7}}{-\tfrac{1}{15} + \tfrac{2}{3}} = \frac{\tfrac{1}{63}}{\tfrac{9}{15}}

\tfrac{1}{63} : \tfrac{9}{15} = \tfrac{1}{63} \cdot \tfrac{15}{9} = \tfrac{5}{189}

Notes in the margin

6^{-1} = \tfrac{1}{6}

\left(\sqrt{3}\right)^{-4} = \left(\tfrac{1}{\sqrt{3}}\right)^{4} = \tfrac{1}{9}

0.5^{-2} = \left(\tfrac{1}{2}\right)^{-2} = 2^{2} = 4

\sqrt[3]{-8^{-1}} = \sqrt[3]{-\tfrac{1}{8}} = \left(-\tfrac{1}{8}\right)^{\tfrac{1}{3}} = -\tfrac{1}{2}

\log_{8} 0.5 = \log_{8} \tfrac{1}{2} = -\tfrac{1}{3} \;\Rightarrow\; 8^{-\tfrac{1}{3}} = \tfrac{1}{2}

\log_{32} 2 = \tfrac{1}{5} \;\Rightarrow\; 32^{\tfrac{1}{5}} = 2

Note that 2⁵ = 32.

\tfrac{1}{6} - \tfrac{1}{9} = \tfrac{3}{18} - \tfrac{2}{18} = \tfrac{1}{18}

4 - \tfrac{1}{2} = 3\tfrac{1}{2} = \tfrac{7}{2}

-\tfrac{1}{15} + \tfrac{2}{3} = -\tfrac{1}{15} + \tfrac{10}{15} = \tfrac{9}{15}

Frequently asked questions

How do you work out a power with a negative exponent?

Take the reciprocal of the base and raise it to the positive exponent. That is why 0.5 to the power of minus two is two squared, which is four.

What is the logarithm base 8 of 0.5?

Minus one third, because eight to the power of minus one third is one half. Write 0.5 as one half and look for the exponent with base eight.

How do you divide by a compound fraction?

Multiply by its reciprocal. Dividing one sixty-third by nine fifteenths means multiplying by fifteen ninths.

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Powers with logarithmic exponents

Values of a rational function at irrational arguments