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# How do you rewrite $5{x^{ - 4}}$ using positive exponents?

Last updated date: 28th Feb 2024
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Hint: We will use the algebraic formula of indices to convert the expression of negative exponents into positive exponents. On doing some simplification we get the required answer.

Formula used: Multiplication rule of Indices:
If we multiply any variable $n$number of times, then we will put the addition of this $n$numbers into the power of that variable.
Like,
$(m \times m \times m \times m \times ............n\;number\;of\;times)$
$= {m^{(1 + 2 + 3 + 4 + ..... + n)}}$.
Division rule of Indices:
If we divide a variable having an exponent to its power by the same variable having a different exponent to its power, then we need to subtract the components in the power.
Like,
$\dfrac{{{a^m}}}{{{a^n}}} = {a^{(m - n)}}\;,\;where\;m,n\;are\;any\;number.$
If $m = 0$, then we can set:
${a^m} = {a^0}$.
Which implies that ${a^0} = 1$.
Now, we can derive the above equation as $\dfrac{{{a^m}}}{{{a^n}}} = \dfrac{1}{{{a^n}}},\;when\;m = 0.$

Complete Step by Step Solution:
We have to find the positive exponent of $5{x^{ - 4}}$.
Let's say, $A = 5{x^{ - 4}}$.
As we know $(0 - 4) = - 4$, we can rewrite the above expression.
Now, we can expand it into ${a^{m - n}}$form in following way:
$A = 5 \times {x^{(0 - 4)}}$, where $5$is the coefficient of the above expression.
So, if we tally this form with the above formula, we can write the following equations:
$\Rightarrow a = x,\;m = 0,\;n = 4.$
So, we can write the above expression in following way:
$\Rightarrow A = 5 \times \dfrac{{{x^0}}}{{{x^4}}}$.
We know that, ${x^0} = 1$.
By putting this value in the above expression, we get:
$\Rightarrow A = 5 \times \dfrac{1}{{{x^4}}}$.
We can rewrite it in following way:
$\Rightarrow A = \dfrac{5}{{{x^4}}}$.
So, in the above expression every value of the exponent is positive.

Therefore, The required positive exponent form of the expression $5{x^{ - 4}}$ is $\dfrac{5}{{{x^4}}}$.

Note: Alternative way to solve:
We can also use the multiplication property of indices.
It is given that $5{x^{ - 4}}$.
Now, we can rewrite $( - 4)$ as $( - 1) + ( - 1) + ( - 1) + ( - 1)$.
So, we can rewrite $5{x^{ - 4}}$ as $5{x^{( - 1) + ( - 1) + ( - 1) + ( - 1)}}$.
By using the multiplication rule of indices, we can write the following expression:
$\Rightarrow 5 \times {x^{ - 1}} \times {x^{ - 1}} \times {x^{ - 1}} \times {x^{ - 1}}$.
We know that, ${a^{ - n}} = \dfrac{1}{{{a^n}}}$.
So, we can rewrite $5 \times {x^{ - 1}} \times {x^{ - 1}} \times {x^{ - 1}} \times {x^{ - 1}}$ as following way:
$\Rightarrow 5 \times \left( {\dfrac{1}{x} \times \dfrac{1}{x} \times \dfrac{1}{x} \times \dfrac{1}{x}} \right)$
By simplifying it, we get:
$\Rightarrow \dfrac{5}{{{x^4}}}$.
So, required positive exponent form $\dfrac{5}{{{x^4}}}$.