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How do you rewrite \[5{x^{ - 4}}\] using positive exponents?

seo-qna
Last updated date: 28th Feb 2024
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IVSAT 2024
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 \[\dfrac{5}{{{x^4}}}\].