Question

How do you simplify \[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}\] and write it using only positive exponents?

Answer 1

Hint: To simplify the given algebraic expression, we have to keep the following in mind: One of the important tools we may use while we simplify an algebraic expression is \[{{x}^{n}}{{x}^{m}}={{x}^{n+m}}.\] If we change the sign of the exponents in the same identity, we get the identity as \[{{x}^{-n}}{{x}^{-m}}={{x}^{-n-m}}={{x}^{-\left( n+m \right)}}.\] We surely know that \[{{\left( {{x}^{n}} \right)}^{m}}={{x}^{nm}}.\] We can see from the above identity that \[{{\left( \dfrac{1}{{{x}^{n}}} \right)}^{m}}=\dfrac{1}{{{x}^{nm}}}.\] Among this list we should also include \[{{x}^{-n}}=\dfrac{1}{{{x}^{n}}}.\]

Complete step by step solution:
Consider the algebraic expression \[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}.\]
To simplify this expression, we should use \[{{x}^{-n}}{{x}^{-m}}={{x}^{-n-m}}={{x}^{-\left( n+m \right)}}.......(1)\]
Let us take \[{{x}^{-2}}{{x}^{-3}}\] into consideration.
By using the equation \[(1),\] we will get \[{{x}^{-2}}{{x}^{-3}}={{x}^{-2-3}}.\]
This now becomes \[{{x}^{-2}}{{x}^{-3}}={{x}^{-2-3}}={{x}^{-5}}.\]
Now consider \[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}.\]
Remember what we do next is applying \[{{\left( {{x}^{n}} \right)}^{m}}={{x}^{nm}}.......(2)\]
Now from \[(2)\] we will get, \[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}={{\left( {{x}^{-2-3}} \right)}^{4}}={{\left( {{x}^{-\left( 2+3 \right)}} \right)}^{4}}={{\left( {{x}^{-5}} \right)}^{4}}\]
From this we get, \[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}={{\left( {{x}^{-5}} \right)}^{4}}={{x}^{\left( -5 \right)4}}={{x}^{-20}}.\]
Now we are going to apply this identity \[{{x}^{-n}}=\dfrac{1}{{{x}^{n}}}.......(3)\] in the above obtained expression.
We use equation \[(3)\] to get \[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}={{x}^{-20}}=\dfrac{1}{{{x}^{20}}}\] which is the simplified form of the given algebraic expression with only positive exponents.

Note: Remember the above given identities while simplifying the algebraic expressions.
We can simplify this algebraic expression in another way using equations \[(1),(2)\] and \[(3):\] Take \[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}.\]
Do what we have done earlier, \[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}={{\left( {{x}^{-2-3}} \right)}^{4}}.\]
Repeat the next step also to get \[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}={{\left( {{x}^{-\left( 2+3 \right)}} \right)}^{4}}={{\left( {{x}^{-5}} \right)}^{4}}\]
Now use equation \[(3)\] and we will get, \[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}={{\left( {{x}^{-5}} \right)}^{4}}={{\left( \dfrac{1}{{{x}^{5}}} \right)}^{4}}\]
We know, \[{{\left( \dfrac{1}{{{x}^{n}}} \right)}^{m}}=\dfrac{1}{{{x}^{nm}}}.......(4)\]
With the help of equation \[(4)\] we write, \[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}={{\left( \dfrac{1}{{{x}^{5}}} \right)}^{4}}=\dfrac{1}{{{x}^{5.4}}}=\dfrac{1}{{{x}^{20}}}.\]
If we determine to write in a different method, we should go with changing the order of applying the above equations/identities like this:
Write the given algebraic expression as,
\[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}={{\left( \dfrac{1}{{{x}^{2}}}\dfrac{1}{{{x}^{3}}} \right)}^{4}},\] using equation \[(3)\].
Since the left-hand side is a product, we can directly multiply the denominators to get,
 \[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}={{\left( \dfrac{1}{{{x}^{2}}}\dfrac{1}{{{x}^{3}}} \right)}^{4}}={{\left( \dfrac{1}{{{x}^{2}}.{{x}^{3}}} \right)}^{4}}\]
This will now give us,
\[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}={{\left( \dfrac{1}{{{x}^{2+3}}} \right)}^{4}},\] here we have used the equation \[{{x}^{n}}{{x}^{m}}={{x}^{n+m}}\]
Then,
\[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}={{\left( \dfrac{1}{{{x}^{5}}} \right)}^{4}}\]
Use equation \[(4)\] and we will obtain,
\[{{\left( {{x}^{-2}}{{x}^{-3}} \right)}^{4}}=\dfrac{1}{{{x}^{5.4}}}=\dfrac{1}{{{x}^{20}}}.\]