Question

How do you simplify \[\dfrac{{{{\left( {{n^{ - 1}}} \right)}^4}}}{{{n^4}}}\] and write it using only positive exponents?

Answer 1

Hint: First multiply the exponents in ${\left( {{n^{ - 1}}} \right)^4}$ by applying the power rule and multiply the exponents. Then, use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents. Next, add the exponents in ${n^4} \times {n^4}$ by applying the product rule and adding the exponents. We will get the simplified version of \[\dfrac{{{{\left( {{n^{ - 1}}} \right)}^4}}}{{{n^4}}}\].

Formula used:
Quotient property of exponents:
If $a$ is a real number, $a \ne 0$, and $m,n$ are whole numbers, then
$\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$, $m > n$ and $\dfrac{{{a^m}}}{{{a^n}}} = \dfrac{1}{{{a^{n - m}}}}$, $n > m$
Negative exponent:
If $n$ is a positive integer and $a \ne 0$, then ${a^{ - n}} = \dfrac{1}{{{a^n}}}$.

Complete step by step answer:
The Quotient Property of Exponents, introduced in Divide Monomials, had two forms depending on whether the exponent in the numerator or denominator was larger.
Quotient property of exponents:
If $a$ is a real number, $a \ne 0$, and $m,n$ are whole numbers, then
$\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$, $m > n$ and $\dfrac{{{a^m}}}{{{a^n}}} = \dfrac{1}{{{a^{n - m}}}}$, $n > m$
Negative exponent:
If $n$ is a positive integer and $a \ne 0$, then ${a^{ - n}} = \dfrac{1}{{{a^n}}}$.
The negative exponent tells us to rewrite the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.
Multiply the exponents in ${\left( {{n^{ - 1}}} \right)^4}$ by applying the power rule and multiplying the exponents, ${\left( {{a^m}} \right)^n} = {a^{mn}}$.
\[ \Rightarrow \dfrac{{{n^{ - \left( {1 \times 4} \right)}}}}{{{n^4}}}\]
Multiply $1$ with $4$.
\[ \Rightarrow \dfrac{{{n^{ - 4}}}}{{{n^4}}}\]
Now, use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.
\[ \Rightarrow \dfrac{1}{{{n^4} \times {n^4}}}\]
Add the exponents in ${n^4} \times {n^4}$ by applying the product rule and adding the exponents, ${a^m} \times {a^n} = {a^{m + n}}$.
\[ \Rightarrow \dfrac{1}{{{n^8}}}\]

Therefore, the simplified version of \[\dfrac{{{{\left( {{n^{ - 1}}} \right)}^4}}}{{{n^4}}}\] is $\dfrac{1}{{{n^8}}}$.

Note: Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form, $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$, where $a \ne 0$ and, $m$and $n$ are integers.
When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents, ${a^{ - n}} = \dfrac{1}{{{a^n}}}$.