Question

If $x={{\log }_{3}}243,y={{\log }_{2}}64$, then the value of $\sqrt{x-2\sqrt{y}}$ is
(a) $\sqrt{5-2\sqrt{6}}$
(b) $2-\sqrt{3}$
(c) $\sqrt{3}-\sqrt{2}$
(d) $\sqrt{3}-4$

Answer 1

Hint:Write the prime factorization of 243 and 64. Use the logarithmic formula ${{\log }_{a}}{{a}^{b}}=b$ to simplify the given expressions and calculate the value of x and y. Substitute these values in the formula $\sqrt{x-2\sqrt{y}}$. Also, use the algebraic identity ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$ to calculate the value of the given expression.

Complete step-by-step answer:
We know that $x={{\log }_{3}}243,y={{\log }_{2}}64$. We have to calculate the value of $\sqrt{x-2\sqrt{y}}$.
We will first write the prime factorization of 243 and 64. Thus, we have $243={{3}^{5}}$ and $64={{2}^{6}}$.
So, we can rewrite $x={{\log }_{3}}243,y={{\log }_{2}}64$ as $x={{\log }_{3}}243={{\log }_{3}}{{3}^{5}}$ and $y={{\log }_{2}}64={{\log }_{2}}{{2}^{6}}$.
We know the logarithmic formula ${{\log }_{a}}{{a}^{b}}=b$.
Substituting $a=3,b=5$ in the above formula, we have ${{\log }_{3}}{{3}^{5}}=5$.
Similarly, substituting $a=2,b=6$ in the above formula, we have ${{\log }_{2}}{{2}^{6}}=6$.
Thus, we have $x={{\log }_{3}}243={{\log }_{3}}{{3}^{5}}=5$ and $y={{\log }_{2}}64={{\log }_{2}}{{2}^{6}}=6$.
We will substitute the above values in the expression $\sqrt{x-2\sqrt{y}}$. So, we have $\sqrt{x-2\sqrt{y}}=\sqrt{5-2\sqrt{6}}$.
We can also simplify the above expression by writing it as $\sqrt{5-2\sqrt{6}}=\sqrt{3+2-2\sqrt{2}\times \sqrt{3}}=\sqrt{{{\left( \sqrt{3} \right)}^{2}}+{{\left( \sqrt{2} \right)}^{2}}-2\sqrt{2}\times \sqrt{3}}.....\left( 1 \right)$.
We know the algebraic identity ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$.
Substituting $a=\sqrt{3},b=\sqrt{2}$ in the above expression, we have ${{\left( \sqrt{3}-\sqrt{2} \right)}^{2}}.....\left( 2 \right)$.
Substituting equation (2) in equation (1), we have $\sqrt{5-2\sqrt{6}}=\sqrt{{{\left( \sqrt{3}-\sqrt{2} \right)}^{2}}}=\sqrt{3}-\sqrt{2}$.
Hence, the possible values of the expression $\sqrt{x-2\sqrt{y}}$ are $\sqrt{5-2\sqrt{6}}$ and $\sqrt{3}-\sqrt{2}$, which are options (a) and (c).

Note: We must simplify the algebraic expression after substituting the values. Otherwise, we will get only one correct answer. One should also know the difference between an algebraic identity and algebraic expression. An algebraic identity is an equality that holds for all possible values of its variables. An algebraic expression differs from an algebraic identity in the way that the value of algebraic expression changes with the change in variables.