Question

Solve the equation for \[x\]: \[{\left(81\right)}^{\dfrac{3}{4}}-{\left(\dfrac{1}{32}\right)}^{\dfrac{-2}{5}}+x{\left(\dfrac{1}{2}\right)}^{-1}.{2^0} = 27\]

Answer 1

Hint: Use the properties of exponents to solve the terms inside the brackets to write it into simpler forms and get smaller terms. Then treat it as a normal equation and solve it by equating it to the right hand side of the equation.

Complete step-by-step solution:
We are given with the equation \[{\left(81\right)}^{\dfrac{3}{4}}-{\left(\dfrac{1}{32}\right)}^{\dfrac{-2}{5}}+x{\left(\dfrac{1}{2}\right)}^{-1}.{2^0} = 27\]. Let us use the properties of exponents to get it into simpler form.
That is, \[{({(81)^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 4$}}}})^3} - {(32)^{{\raise0.5ex\hbox{$\scriptstyle 2$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 5$}}}} + x{(2)^1}{.2^0} = 27\]
As we know \[{x^{ - a}} = {(\dfrac{1}{x})^a},{x^{ab}} = {({x^a})^b}\]and using the properties of exponents.
Then, \[{({(81)^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 4$}}}})^3} - {({(32)^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 5$}}}})^2} + x{(2)^1}{.2^0} = 27\]
We also know that \[{a^0} = 1 \Rightarrow {2^0} = 1\]
And \[{(81)^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 4$}}}} = 3,{\kern 1pt} {\kern 1pt} {\kern 1pt} {(32)^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 5$}}}} = 2\]then substituting these values into the equation we get,
\[{(3)^3} - {(2)^2} + x{(2)^1}.1 = 27\]
Now simplifying the equation further we get,
\[27 - 4 + 2x = 27\]
\[ \Rightarrow 27 - 4 + 2x - 27 = 0\]
\[ \Rightarrow 2x - 4 = 0\]
Now taking common terms from the obtained equation we get,
\[ \Rightarrow 2(x - 2) = 0\]
Now equating it to right hand side we get,
\[ \Rightarrow x = 2\]
Therefore we reached our final solution for the equation i.e. \[x = 2\].
Additional information: Exponential notation is a form of mathematical shorthand which permits us to put in writing complicated expressions extra succinctly. An exponent is a quantity or letter written above and to the right of the number in a mathematical expression referred to as the base. It shows that the base is to be raised to a certain power.\[x\] is the base and \[n\] is the exponent. That is: If \[x\] is a positive quantity and \[n\] is its exponent, then \[{x^n}\]implies \[x\] is extended by way of itself \[n\] times.

Note: It is important that we know the proper use of the properties of the exponents and its basic definition. It is also advisable that we know the basic results of the exponents beforehand that gives us an advantage of doing the problem faster and within a less amount of time.