Question

Ascending order of \[ a=7^{\frac23},\; b=125^{\frac12},\; c=135^{\frac12},\; d=59^{\frac13} \] is

• A. \(a,b,d,c\)
• B. \(a,c,b,d\)
• C. \(a,d,b,c\)
• D. \(a,d,c,b\)

Hint:

For comparing surds and fractional powers, convert into decimal approximations for easy ordering.

Answer 1

The Correct Option is C

Concept: To compare numbers with powers, convert them into approximate values.

Step 1: Find value of \(a\).
\[ a=7^{\frac23}=(\sqrt[3]{7})^2 \] \[ \sqrt[3]{7}\approx 1.91 \] \[ a\approx (1.91)^2=3.65 \]

Step 2: Find value of \(d\).
\[ d=59^{\frac13} \] \[ \approx 3.89 \]

Step 3: Find value of \(b\).
\[ b=125^{\frac12} \] \[ =\sqrt{125} \] \[ =5\sqrt5\approx 11.18 \]

Step 4: Find value of \(c\).
\[ c=135^{\frac12} \] \[ =\sqrt{135} \] \[ =3\sqrt{15}\approx 11.62 \]

Step 5: Arrange in ascending order.
\[ 3.65<3.89<11.18<11.62 \] Thus: \[ a<d<b<c \] Hence, the ascending order is: \[ \boxed{a,d,b,c} \]