Question

How do you solve \[{{36}^{2p}}={{216}^{p-1}}\]?

Answer 1

Hint: This type of problem can be solved using power rule of multiplication, that is, if \[{{x}^{a}}={{x}^{b}}\], then a=b. First, we have to consider the given equation with variable p. And simplify the given function using the power rule of multiplication in such a way that the base of LHS and RHS is the same. Here, we find that \[{{6}^{2}}=36\] and \[{{6}^{3}}=216\] . Then, use the rule \[{{\left( {{a}^{n}} \right)}^{m}}={{a}^{nm}}\] and simplify the equation so that the base of the equation is 6. And equate the power where we get a linear equation with one variable p. On solving p, we get the final answer.

Complete step-by-step solution:
According to the question, we are asked to solve \[{{36}^{2p}}={{216}^{p-1}}\].
We have been given the equation is \[{{36}^{2p}}={{216}^{p-1}}\]. ---------(1)
We have to first convert both the bases of LHS and RHS into the same number.
Let us first consider LHS.
LHS=\[{{36}^{2p}}\]
LHS= \[{{\left( 36 \right)}^{2p}}\]
We know that \[{{6}^{2}}=36\]. Using this in LHS, we get
\[{{\left( 36 \right)}^{2p}}={{\left( {{6}^{2}} \right)}^{2p}}\].
We know that \[{{\left( {{a}^{n}} \right)}^{m}}={{a}^{nm}}\]. Therefore, using this property, we get
\[{{\left( 36 \right)}^{2p}}={{6}^{2\times 2p}}\]
\[\Rightarrow {{\left( 36 \right)}^{2p}}={{6}^{4p}}\]
Therefore, LHS= \[{{6}^{4p}}\].
Now consider the RHS.
RHS= \[{{216}^{p-1}}\]
RHS=\[{{\left( 216 \right)}^{p-1}}\]
We know that \[{{6}^{3}}=216\]. Using this in the RHS, we get
We know that \[{{\left( {{a}^{n}} \right)}^{m}}={{a}^{nm}}\]. Therefore, using this property, we get
\[{{\left( 216 \right)}^{p-1}}={{\left( 6 \right)}^{3\left( p-1 \right)}}\]
\[\Rightarrow {{\left( 216 \right)}^{p-1}}={{6}^{3\left( p-1 \right)}}\]
Therefore, RHS= \[{{6}^{3\left( p-1 \right)}}\].
Equating LHS and RHS, we get
\[{{6}^{4p}}={{6}^{3\left( p-1 \right)}}\]
Now, we have to use the property “if \[{{x}^{a}}={{x}^{b}}\], then a=b”.
Here, $x=6, a=4p$ and $b=3(p-1)$.
Therefore, $4p=3(p-1)$.
Using the distributive property, that is, \[a\left( b+c \right)=ab+ac\], we get
$4p=3p-3$
Now, subtract 3p from both the sides of the equation.
$4p-3p=3p-3p-3$
On further simplification, we get
$p=-3$
Therefore, the value of p in the equation \[{{36}^{2p}}={{216}^{p-1}}\] is -3.

Note: We should know the properties of power to solve this type of problem. Also avoid calculation mistakes based on sign conventions. We should use the properties of power to solve these types of questions without which we cannot find the final answer. Similarly, we can simplify the functions with two variables also.