Mathematics Questions and Answers

Simplify (\(\frac{1}{x^{-1}} + \frac{1}{y^{-1}}\))^-1 = (\(\frac{1}{x^{-1}} + \frac{1}{y^{-1}}\))^-1

= (x + y)^-1 = \(\frac{(x)}{y}\)

= \(\frac{x}{y}\)

3 log₆9 + log₆12 + log₆64 - log₆72

= log₆9³ + log₆12 + log₆64 - log₆72

log₆729 + log₆12 + log₆64 - log₆72

log₆(729 x 12 x 64) = log₆776

= log₆6₅ = 5 log₆6 = 5

N.B: log₆6 = 1

\(\sqrt{27}\) + \(\frac{3}{\sqrt{3}}\)

= \(\sqrt{9 \times 3}\) + \(\frac{3 \times {\sqrt{3}}}{{\sqrt{3}} \times {\sqrt{3}}}\)

= 3\(\sqrt{3}\) + \(\sqrt{3}\)

= 4\(\sqrt{3}\)

\(\sqrt{160r^2 + \sqrt{71r^4 + \sqrt{100r^8}}}\)

Simplifying from the innermost radical and progressing outwards we have the given expression

\(\sqrt{160r^2 + \sqrt{71r^4 + \sqrt{100r^8}}}\) = \(\sqrt{160r^2 + \sqrt{71r^4 + 10r^4}}\)

= \(\sqrt{160r^2 + \sqrt{81r^4}}\)

\(\sqrt{160r^2 + 9r^2}\) = \(\sqrt{169r^2}\)

= 13r

use "T" to represent the total profit. The first receives \(\frac{1}{3}\) T

remaining, 1 - \(\frac{1}{3}\)

= \(\frac{2}{3}\)T

The seconds receives the remaining, which is \(\frac{2}{3}\) also

\(\frac{2}{3}\) x \(\frac{2}{3}\) = \(\frac{4}{9}\)

The third receives the left over, which is \(\frac{2}{3}\)T - \(\frac{4}{9}\)T = (\(\frac{6 - 4}{9}\))T

= \(\frac{2}{9}\)T

The third receives \(\frac{2}{9}\)T which is equivalent to N12000

If \(\frac{2}{9}\)T = N12, 000

T = \(\frac{12 000}{\frac{2}{9}}\)

Total share[T] = N54, 000

The first receives \(\frac{1}{3}\) of T → \(\frac{1}{3}\) * N54, 000 = N18,000

The second receives \(\frac{4}{9}\) of T → \(\frac{4}{9}\) * N54, 000 = N24,000

The third receives \(\frac{2}{9}\)T which is equivalent to N12000.

Adding the three shares give total profit of N54,000