Question

Suppose a, b are positive real numbers such that, $a\sqrt a + b\sqrt b = 183$, $a\sqrt b + b\sqrt a = 182.$ Find $\dfrac{9}{5}(a + b)$.

Answer 1

Hint: According to given in the question a, and b are real numbers and to find the value of $\dfrac{9}{5}(a + b)$. We have to solve the given expressions $a\sqrt a + b\sqrt b = 183$ and $a\sqrt b + b\sqrt a = 182.$ Hence, to solve the given expressions first of all we have to rearrange the terms of $a\sqrt a + b\sqrt b = 183$ and $a\sqrt b + b\sqrt a = 182$.
So, in the given expression we can write $a$ in the form of ${(\sqrt a )^2}$ and same as we can write $b$ in the form of ${(\sqrt b )^2}$
Now, to solve the expression after rearranging it we have to let $\sqrt a $ as $m$ and $\sqrt b $ as $n$.
Hence, on solving we will obtain two different expressions and now we will convert both of the two different equations in the form of a single equation. After converting in the form of a single equation we will try to make that in the form of the formula which is given below:

Formula used:
${(x + y)^3} = {x^3} + {y^3} + 3xy(x + y).............................(1)$
$({x^3} + {y^3}) = (x + y)({x^2} + {y^2} - xy)..............................(a)$
Therefore, on making the obtained equation in the form of the whole cube we can find the sum of the two real numbers a and b then we can substitute the sum in the equation $\dfrac{9}{5}(a + b)$ to obtain the value.

Complete step by step answer:
Given,
$a\sqrt a + b\sqrt b = 183$ and,
$a\sqrt b + b\sqrt a = 182$ where, $a$ and $b$ are real numbers.
Step 1: First of all we have to rearrange the given expressions $a\sqrt a + b\sqrt b = 183$ and $a\sqrt b + b\sqrt a = 182.$
As we know that, we can write $a$ in the form of ${(\sqrt a )^2}$ and same as we can write $b$ in the form of ${(\sqrt b )^2}$
Hence, on rearranging the terms of $a\sqrt a + b\sqrt b = 183$ the obtained equation is:
$\Rightarrow {(\sqrt a )^2}\sqrt a + {(\sqrt b )^2}\sqrt b = 183$
On solving the obtained expression,
$\Rightarrow {(\sqrt a )^3} + {(\sqrt b )^3} = 183$……………………………..(2)
Step 2: Now, we will rearrange the terms of the expression $a\sqrt b + b\sqrt a = 182.$ As we know that, we can write $a$ in the form of ${(\sqrt a )^2}$ and same as we can write $b$ in the form of ${(\sqrt b )^2}$
Hence, on rearranging the terms of $a\sqrt b + b\sqrt a = 182$ the obtained equation is:
$\Rightarrow {(\sqrt a )^2}\sqrt b + {(\sqrt b )^2}\sqrt a = 182$……………………...(3)
Step 3: Now, to solve the obtained expressions (2) and (3) we will let that $\sqrt a $ is $m$ and $\sqrt b $ is $n$.
Hence,
$\Rightarrow \sqrt a = m$ and, $\sqrt b = n$
On substituting in the equation (2),
$\Rightarrow {m^3} + {n^3} = 183.........................(4)$
Same as on substituting in the equation (3),
$\Rightarrow {m^2}n + {n^2}m = 182.......................(5)$
Step 4: On rearranging the terms of equation (5) obtained just above,
\[\Rightarrow mn(m + n) = 182.......................(6)\]
Step 5: Multiplying with 3 on the both sides of the equation (6)
$3mn(m + n) = 3 \times 182$………………(7)
Now, on adding the obtained equations (4) and (7)
$
\Rightarrow {m^3} + {n^3} + 3mn(m + n) = 183 + 546 \\
\Rightarrow {m^3} + {n^3} + 3mn(m + n) = 729....................(8) \\
 $
Step 6: Now, to solve the equation (8) as obtained just above we will use the formula (1) as mentioned in the solution hint.
$\Rightarrow {m^3} + {n^3} + 3mn(m + n) = {(m + n)^3}$…………………(9)
Hence, on substituting the expression (8) in the obtained expression (9)
$\Rightarrow {(m + n)^3} = 729$
On solving the obtained expression,
$
\Rightarrow (m + n) = \sqrt[3]{{729}} \\
\Rightarrow (m + n) = \sqrt[3]{{9 \times 9 \times 9}} \\
\Rightarrow (m + n) = 9 \\
 $
Step 7: Now, we will again solve the expression (4) with the help of the formula (a) as mentioned in the solution hint.
$\Rightarrow {m^3} + {n^3} = 183$
Now, as we know
$\Rightarrow {m^3} + {n^3} = (m + n)({m^2} + {n^2} - mn)$
Hence,
$\Rightarrow (m + n)({m^2} + {n^2} - mn) = 183$
On rearranging the terms of obtained expression,
$\Rightarrow (m + n)({m^2} + {n^2}) - mn(m + n) = 183 $
On substituting the equation (6) in the equation obtained just above,
$
\Rightarrow (m + n)({m^2} + {n^2}) - 182 = 183 \\
\Rightarrow (m + n)({m^2} + {n^2}) = 365 \\
\Rightarrow ({m^2} + {n^2}) = \dfrac{{365}}{{(m + n)}} \\
 $
As we have already obtained the value of $(m + n)$hence, on substituting in the equation obtained just above,
$\Rightarrow \dfrac{{365}}{{(m + n)}} = \dfrac{{635}}{9}$
Therefore, the required expression is:
$
\Rightarrow \dfrac{9}{5}(m + n) = \dfrac{9}{5}({m^2} + {n^2}) \\
   = \dfrac{9}{5} \times \dfrac{{365}}{9} \\
   = 73 \\
 $

Hence, if a, b are positive real numbers such that $a\sqrt a + b\sqrt b = 183$, $a\sqrt b + b\sqrt a = 182.$ Then value of $\dfrac{9}{5}(a + b) = 73$

Note:
To solve the given expression we can write $a$ in the form of ${(\sqrt a )^2}$ and same as we can write $b$ in the form of ${(\sqrt b )^2}$ so that we can easily solve the equation by converting it.
To solve these types of expressions we should always try to make a single expression by solving the expressions given in the questions.
Be careful while substituting the values as obtained during solving the expressions as given in the question.