Soalan 2:
Lengkapkan rajah berikut dengan nilai yang betul.
Penyelesaian:
(a)
$$ \begin{aligned} \sqrt[y]{6561^x} & =81 \\ 6561^{\frac{x}{y}} & =3^4 \\ \left(3^8\right)^{\frac{x}{y}} & =3^4 \\ (3)^{\frac{8 x}{y}} & =3^4 \\ \frac{8 x}{y} & =4 \\ \frac{x}{y} & =\frac{4}{8} \\ \frac{x}{y} & =\frac{1}{2} \\ x & =1, y=2 \\ \therefore \sqrt[2]{6561^1} & =81 \end{aligned} $$
$$ \begin{aligned} 3^x & =81 \\ 3^x & =3^4 \\ x & =4 \\ \therefore 3^4 & =81 \end{aligned} $$
$$ \begin{aligned} 9^x & =81 \\ 9^x & =9^2 \\ x & =2 \\ \therefore 9^2 & =81 \end{aligned} $$
$$
\begin{aligned}
81^x & =81 \\
81^x & =81^1 \\
x & =1 \\
\therefore 81^1 & =81
\end{aligned}
$$
$$ \begin{aligned} 243^{\frac{4}{x}} & =81 \\ \left(3^5\right)^{\frac{4}{x}} & =3^4 \\ (3)^{\frac{20}{x}} & =3^4 \\ \frac{20}{x} & =4 \\ x & =\frac{20}{4} \\ x & =5 \\ \therefore 243^{\frac{4}{5}} & =81 \end{aligned} $$
$$ \begin{aligned} 27^{\frac{x}{3}} & =81 \\ \left(3^8\right)^{\frac{x}{3}} & =3^4 \\ 3^x & =3^4 \\ x & =4 \\ \therefore 27^{\frac{4}{3}} & =81 \end{aligned} $$
(b)
$$ \begin{aligned} 25^{\frac{3}{x}} & =125 \\ \left(5^2\right)^{\frac{3}{x}} & =5^3 \\ (5)^{\frac{6}{x}} & =5^3 \\ \frac{6}{x} & =3 \\ x & =\frac{6}{3} \\ x & =2 \\ \therefore 25^{\frac{3}{2}} & =125 \end{aligned} $$
$$ \begin{aligned} 125^x & =125 \\ 125^x & =125^1 \\ x & =1 \\ \therefore 125^1 & =125 \end{aligned} $$
$$ \begin{aligned} 625^{\frac{x}{4}} & =125 \\ \left(5^4\right)^{\frac{x}{x}} & =5^3 \\ 5^x & =5^3 \\ x & =3 \\ \therefore 625^{\frac{3}{4}} & =125 \end{aligned} $$
$$ \begin{aligned} \sqrt[y]{15625^x} & =125 \\ 15625^{\frac{x}{y}} & =5^3 \\ \left(5^6\right)^{\frac{x}{y}} & =5^3 \\ (5)^{\frac{6 x}{y}} & =5^3 \\ \frac{6 x}{y} & =3 \\ \frac{x}{y} & =\frac{3}{6} \\ \frac{x}{y} & =\frac{1}{2} \\ x=1, y & =2 \\ \therefore \sqrt[2]{15625^1} & =125 \end{aligned} $$
$$ \begin{aligned} 3125^{\frac{3}{x}} & =125 \\ \left(5^5\right)^{\frac{3}{x}} & =5^3 \\ \frac{15}{x} & =3 \\ x & =5 \\ \therefore 3125^{\frac{3}{5}} & =125 \end{aligned} $$
$$ \begin{aligned} 5^x & =125 \\ 5^x & =5^3 \\ x & =3 \\ \therefore 5^3 & =125 \end{aligned} $$
Lengkapkan rajah berikut dengan nilai yang betul.
Penyelesaian:
(a)
$$ \begin{aligned} \sqrt[y]{6561^x} & =81 \\ 6561^{\frac{x}{y}} & =3^4 \\ \left(3^8\right)^{\frac{x}{y}} & =3^4 \\ (3)^{\frac{8 x}{y}} & =3^4 \\ \frac{8 x}{y} & =4 \\ \frac{x}{y} & =\frac{4}{8} \\ \frac{x}{y} & =\frac{1}{2} \\ x & =1, y=2 \\ \therefore \sqrt[2]{6561^1} & =81 \end{aligned} $$
$$ \begin{aligned} 3^x & =81 \\ 3^x & =3^4 \\ x & =4 \\ \therefore 3^4 & =81 \end{aligned} $$
$$ \begin{aligned} 9^x & =81 \\ 9^x & =9^2 \\ x & =2 \\ \therefore 9^2 & =81 \end{aligned} $$
$$
\begin{aligned}
81^x & =81 \\
81^x & =81^1 \\
x & =1 \\
\therefore 81^1 & =81
\end{aligned}
$$$$ \begin{aligned} 243^{\frac{4}{x}} & =81 \\ \left(3^5\right)^{\frac{4}{x}} & =3^4 \\ (3)^{\frac{20}{x}} & =3^4 \\ \frac{20}{x} & =4 \\ x & =\frac{20}{4} \\ x & =5 \\ \therefore 243^{\frac{4}{5}} & =81 \end{aligned} $$
$$ \begin{aligned} 27^{\frac{x}{3}} & =81 \\ \left(3^8\right)^{\frac{x}{3}} & =3^4 \\ 3^x & =3^4 \\ x & =4 \\ \therefore 27^{\frac{4}{3}} & =81 \end{aligned} $$
(b)
$$ \begin{aligned} 25^{\frac{3}{x}} & =125 \\ \left(5^2\right)^{\frac{3}{x}} & =5^3 \\ (5)^{\frac{6}{x}} & =5^3 \\ \frac{6}{x} & =3 \\ x & =\frac{6}{3} \\ x & =2 \\ \therefore 25^{\frac{3}{2}} & =125 \end{aligned} $$
$$ \begin{aligned} 125^x & =125 \\ 125^x & =125^1 \\ x & =1 \\ \therefore 125^1 & =125 \end{aligned} $$
$$ \begin{aligned} 625^{\frac{x}{4}} & =125 \\ \left(5^4\right)^{\frac{x}{x}} & =5^3 \\ 5^x & =5^3 \\ x & =3 \\ \therefore 625^{\frac{3}{4}} & =125 \end{aligned} $$
$$ \begin{aligned} \sqrt[y]{15625^x} & =125 \\ 15625^{\frac{x}{y}} & =5^3 \\ \left(5^6\right)^{\frac{x}{y}} & =5^3 \\ (5)^{\frac{6 x}{y}} & =5^3 \\ \frac{6 x}{y} & =3 \\ \frac{x}{y} & =\frac{3}{6} \\ \frac{x}{y} & =\frac{1}{2} \\ x=1, y & =2 \\ \therefore \sqrt[2]{15625^1} & =125 \end{aligned} $$
$$ \begin{aligned} 3125^{\frac{3}{x}} & =125 \\ \left(5^5\right)^{\frac{3}{x}} & =5^3 \\ \frac{15}{x} & =3 \\ x & =5 \\ \therefore 3125^{\frac{3}{5}} & =125 \end{aligned} $$
$$ \begin{aligned} 5^x & =125 \\ 5^x & =5^3 \\ x & =3 \\ \therefore 5^3 & =125 \end{aligned} $$