Express each of the following as a single fraction involving positive exponents

zi2lalZ 2021-10-04 Answered
Express each of the following as a single fraction involving positive exponents only.
\(\displaystyle{3}{a}^{{-{2}}}+{4}{b}^{{-{1}}}\)

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Expert Answer

broliY
Answered 2021-10-05 Author has 8393 answers
Step 1
The negative exponents can be converted to positive exponents by taking the reciprocal of the respective term.
We can use the format \(\displaystyle{x}^{{-{n}}}={\frac{{{1}}}{{{x}^{{{n}}}}}}\) to convert the negative exponents to positive exponents.
We can write \(\displaystyle{x}^{{-{2}}}\) as \(\displaystyle{\frac{{{1}}}{{{x}^{{{2}}}}}}\) using the format \(\displaystyle{x}^{{{n}}}={\frac{{{1}}}{{{x}^{{{n}}}}}}\)
Step 2
Consider the expression \(\displaystyle{3}{a}^{{-{2}}}+{4}{b}^{{-{1}}}\)
first we have to convert the negative exponents in the expression to positive exponents by using the form \(\displaystyle{x}^{{-{n}}}={\frac{{{1}}}{{{x}^{{{n}}}}}}\)
The we get the expression as
\(\displaystyle{3}{a}^{{-{2}}}+{4}{b}^{{-{1}}}={3}\times{\frac{{{1}}}{{{a}^{{{2}}}}}}+{4}\times{\frac{{{1}}}{{{b}}}}\)
\(\displaystyle{3}{a}^{{-{2}}}+{4}{b}^{{-{1}}}={\frac{{{3}}}{{{a}^{{{2}}}}}}+{\frac{{{4}}}{{{b}}}}\)
Now we have to solve the fractions to get the single fractional form. We have to find the LCM(LCD) of the denominators of the fractions and we need to convert both the denominators to the LCM value by multiplying with the constant number or expression.
Here we have LCM(least common multiple) of \(\displaystyle{a}^{{{2}}}\) and \(\displaystyle{b}\) is \(\displaystyle{a}^{{{2}}}{b}\), we have to convert both denominators of fractions to \(\displaystyle{a}^{{{2}}}{b}\)
Thus we get
\(\displaystyle{\frac{{{3}}}{{{a}^{{{2}}}}}}+{\frac{{{4}}}{{{b}}}}={\frac{{{3}}}{{{a}^{{{2}}}}}}\times{\frac{{{b}}}{{{b}}}}+{\frac{{{4}}}{{{b}}}}\times{\frac{{{a}^{{{2}}}}}{{{a}^{{{2}}}}}}\)
\(\displaystyle={\frac{{{3}{b}}}{{{a}^{{{2}}}{b}}}}+{\frac{{{4}{a}^{{{2}}}}}{{{a}^{{{2}}}{b}}}}\)
\(\displaystyle={\frac{{{3}{b}+{4}{a}^{{{2}}}}}{{{a}^{{{2}}}{b}}}}\)
Hence we have the single fractional form \(\displaystyle{\frac{{{4}{a}^{{{2}}}+{3}{b}}}{{{a}^{{{2}}}{b}}}}\)
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