Chapter 6.5, Problem 64HP

To determine

To Find :

Find a number other than 1 that has a positive whole number for a square root, cube root, and fourth root.

Answer to Problem 64HP

4096

Explanation of Solution

Concept Used :

• Definition of nth root:
• Words : For any real numbers a and b , and any positive integer n , if $a^n=b$ then a is the nth root of b . That is $a=\sqrt[n]{b}$
• $\begin{array}{l}\sqrt[n]{b}\\ Here\\ b=radicand\\ n=index\\ \sqrt{}=radicalsign\end{array}$

Calculation:

$\begin{array}{l}\text{We need to find a number whose square root , cube root and fourth root}\\ \text{is positive whole number}\text{.}\\ \text{So, we first take LCM of 2,3,4 =12}\\ \text{And now, take exponent of any whole number as 12}\text{.}\\ {\text{Let us consider 2}}^{12}\\ 2^{12}=4096\\ \sqrt{4096}=\sqrt{{(2^6)}^2}=2^6=64\\ \sqrt[3]{4096}=\sqrt[3]{{(2^4)}^3}=2^4=16\\ \sqrt[4]{4096}=\sqrt[4]{{(2^3)}^4}=2^3=8\end{array}$

So, the whole number can be 4096.