Answer

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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 \[\dfrac{5}{{{x^4}}}\].

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