Question

If $\left( 3p-\dfrac{1}{3p} \right)=5$ and $\left( 3p+\dfrac{1}{3p} \right)=x$, then, find the value of $\left( 9{{p}^{2}}-\dfrac{1}{9{{p}^{2}}} \right)$.

Answer 1

Hint: Algebraic Expression is an expression which consists of various variables and constants. Algebraic expression also consists of various algebraic operations and identities. In addition to this, the algebraic expression $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$.

Formula used:
$\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$ identity is used.

Complete step-by-step answer:
Given that,
$\left( 3p-\dfrac{1}{3p} \right)=5$ and $\left( 3p+\dfrac{1}{3p} \right)=x$
Now, using the identity $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$.
$\begin{align}
&\Rightarrow \left( 3p-\dfrac{1}{3p} \right)\left( 3p+\dfrac{1}{3p} \right)=\left( {{\left( 3p \right)}^{2}}-{{\left( \dfrac{1}{3p} \right)}^{2}} \right) \\
&\Rightarrow \left( 3p-\dfrac{1}{3p} \right)\left( 3p+\dfrac{1}{3p} \right)=\left( \left( 9{{p}^{2}} \right)-\left( \dfrac{1}{9{{p}^{2}}} \right) \right) \\
\end{align}$
Put the value of $\left( 3p-\dfrac{1}{3p} \right)=5$ and $\left( 3p+\dfrac{1}{3p} \right)=x$
$\Rightarrow$ $5x=\left( 9{{p}^{2}}-\dfrac{1}{9{{p}^{2}}} \right)$
Hence, the value of $\left( 9{{p}^{2}}-\dfrac{1}{9{{p}^{2}}} \right)=5x$

Additional information:
Various identities of the algebraic expressions are.
Addition identity: ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$
Subtraction identity: ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$

Note: Knowledge about algebraic expressions and its identities is required in order to solve these types of questions. Students can make mistake by squaring the expression as whole instead to individually i.e. $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$ identity can be mistaken for other algebraic identities.