If \(5^x=6^y=30^7\), then what is the value of \(\frac{1}{x}+\frac{1}{y}\)?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\frac{1}{7}\)
Write 5 and 6 as powers of 30. Since \(5\times6=30\), multiplying the two results gives a power of 30 equal to 30 itself, and equal powers of the same base have equal exponents.
5 and 6 as powers of 30:
\(5^{x}=30^{7}\), so \(5=30^{\frac{7}{x}}\) (1)
\(6^{y}=30^{7}\), so \(6=30^{\frac{7}{y}}\) (2)
Multiplying (1) and (2) (the exponents add):
\(5\times6=30^{\frac{7}{x}+\frac{7}{y}}\), that is, \(30^{1}=30^{7\left(\frac{1}{x}+\frac{1}{y}\right)}\)
\(7\left(\frac{1}{x}+\frac{1}{y}\right)=1\)
\(\displaystyle\frac{1}{x}+\frac{1}{y}=\frac{1}{7}\)