## What is Power Operation?

Power operation is topic of algebra in Math. It is use to represent repeated multiplication. Very big number and very small number can be easily express using power operation. Power operation is useful in many fields. In space engineering, it helps in representing the distance or size of particular heavenly body. In medical field, it is used to represent very small size. In medical field it helps to mention size of bacteria or virus.

## Definition

Power is an expression that shows repeated multiplication of the number. The number is called as base and number of times the base is multiplied is called as exponent.

## What is the Difference Between Power and Exponent?

Power is combination of base and exponent. The exponent is the number of times that base is multiplied.

Power can be expressed as a^{b}, here ‘a’ is base and ‘b’ is the exponent. It is read as “a raised to b”. ab means ‘a’ is multiplying by b times. The power of the number can be positive or negative. Zero exponents of any number give the answer 1.

If the power of the number is 2, it read as squared. i.e a^{2} ‘a squared’ If the power of the number is 3 it is called cubed. i.e a^{3}’ a cubed’. If the exponent of the power is negative

then take reciprocal of power and exponent becomes positive. i.e ${a}^{-m}=\frac{1}{{a}^{m}}$

## Rules of Power

- Product rule - The product of two powers with same base and different exponents is equal to same base with sum of the exponents.

Product of a^{m} and a^{n} is a is multiplied by m times and n times. The resultant exponent is ‘m+n’. So the result is a^{m+n} which is a is multiplied by m+n times.

$\begin{array}{c}{a}^{m}\times {a}^{n}=a\times a\times \mathrm{...}\left(m\text{times}\right)\times a\times a\times \mathrm{...}\left(m\text{times}\right)\\ =a\times a\times \mathrm{...}\left(m+n\text{times}\right)\\ ={a}^{m+n}\end{array}$

Here, base is common i.e a and exponents are m and n.

Example –

1. $\begin{array}{c}{3}^{2}\times {3}^{8}={3}^{2+8}\\ ={3}^{10}\end{array}$

2. $\begin{array}{c}{5}^{4}\times {5}^{-2}={5}^{4+\left(-2\right)}\\ ={5}^{2}\end{array}$

2. Division rule - The product of two powers with the same base and different exponents is equal to the same base with subtraction of the exponents. The resultant exponent is ‘m-n’

${a}^{m}\xf7{a}^{n}={a}^{m-n}$

Here,

- if m>n ‘a’ in denominator get cancelled and result gives ‘a’ in numerator so denominator becomes 1.

$\begin{array}{c}\frac{{5}^{5}}{{5}^{3}}={5}^{5-3}\\ ={5}^{2}\end{array}$

2. if m<n ‘a’ in numerator get cancelled and result gives ‘a’ in denominator so numerator becomes 1.

Example –

$\begin{array}{c}\frac{{6}^{3}}{{6}^{7}}={6}^{3-7}\\ ={6}^{-4}\\ =\frac{1}{{6}^{4}}\end{array}$

- Power of a product – if the base is in form of product of two factors then exponent is distributes over base.

If ab is the base and m is the exponent. ‘a’ and ‘b’ are factor of ‘ab’ so ‘m’ is

distributed over ‘a’ and ‘b’ i.e ${\left(ab\right)}^{m}={a}^{m}{b}^{m}$

Example –

${\left(12\right)}^{5}={3}^{5}\cdot {4}^{5}$

4. Power of quotient - if the base is in form of fraction then exponent is distributes over numerator and denominator.

If $\frac{a}{b}$ is the base and m is the exponent. ‘a’ is numerator and ‘b’ is denominator so ‘m’ is distributed over ‘a’ and ‘b’

${\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}}$

- Power one – any number with exponent one is equal to the number itself.

If ‘a’ is the base and one is the exponent. Then

${a}^{1}=a$

- Power zero -If exponent is zero when base is not equal to zero then it is always 1.

If ‘a’ is non-zero number and exponent is zero. i.e a^{0}=1

Example –

${1234}^{0}=1$

7. Negative exponent – if the exponent is negative it gives positive reciprocal.

In ${a}^{-m}$’a’ is base and ‘-m’ is exponent. So, the result is ${a}^{-m}=\frac{1}{{a}^{m}}$.

Example - ${4}^{-2}=\frac{1}{{4}^{2}}$

8. Fractional power – if exponent can be in fractional form.

If exponent is $\frac{1}{2}$then it gives square root of the number.

i.e ${a}^{\frac{1}{2}}=\sqrt{a}$

If exponent is $\frac{1}{3}$then it gives cube root of the number.

i.e ${a}^{\frac{1}{3}}=\sqrt[3]{a}$

in general, If exponent is $\frac{1}{n}$then it gives nth root of the number.

i.e ${a}^{\frac{1}{n}}=\sqrt[n]{a}$

9. Power of power – the power of power gives the result the power is product of both exponents. If power is in parentheses and there is exponent of parentheses as well. The result of such power is product of inner exponent and outer exponent.

Example – in ${\left({a}^{m}\right)}^{n}$here in parentheses${a}^{m}$ base is ‘a’ and exponent is m. for exponent ‘n’, ${a}^{m}$ is the base. So the resultant exponent is mn.${\left({a}^{m}\right)}^{n}={a}^{mn}$

10. In PEMDAS power rules are applied after solving parenthesis in the polynomial.

Example - $\left(3{x}^{2}\right)+{\left(4x\right)}^{2}$

Here we can use power rule

$\begin{array}{c}\left(3{x}^{2}\right)+{\left(4x\right)}^{2}=3{x}^{2}+16{x}^{2}\\ =19{x}^{2}\end{array}$

## Use of Power Operation

- Scientific notation - Power operation is use to represent large numbers or very small numbers. 1000000 can be represent as 10
^{6}. Negative exponent represent smaller value. This is helpful in representing scientific terms. While explaining power in scientific notation the base should be 10. The exponents of the scientific notation is either positive or negative. The exponents of the scientific notation is always non-zero. - If the power is positive then the number is big. In positive exponent decimal point will move on right side.

Example – In scientific notation $34\times {10}^{6}$here 34 is the coefficient and 10 is base and 6 is the exponent. Decimal point will move 6 places on right. It gives the value $34000000$

- If power is negative then the number is smaller. In negative exponent decimal point will move on left side.Example – In scientific notation $4\times {10}^{-6}$here 4 is the coefficient and 10 is base and -6 is the exponent. Decimal point will move 6 places on left. It gives the value $0.000004$

## Example of Power Operation in Real Life

- Speed of the light in vacuum is 300,000 km/sec it can be represent as$3\times {10}^{5}km/\mathrm{sec}$

- The distance between Earth and Sun is $9.3\times {10}^{7}miles$

- Mass of a dust particle is $7.53\times {10}^{-10}kg$

- The distance travel by light in one year is $9.46\times {10}^{15}\text{km}$

## Practice Questions

- Simplify following

Use product rule-

$\begin{array}{c}{y}^{4}{y}^{3}={y}^{4+3}\\ ={y}^{12}\end{array}$

- ${\left(5\right)}^{3}{\left(2\right)}^{3}$

$\begin{array}{c}{\left(5\right)}^{3}{\left(2\right)}^{3}={10}^{3}\\ =1000\end{array}$

2. Convert the scientific notation in decimal notation $3.6\times {10}^{-5}$Here, exponent of base 10 is negative. So decimal point will move 5 places on left. $3.6\times {10}^{-5}=0.000036$

3. Convert the scientific notation in decimal notation $1.24\times {10}^{6}$Here exponent of base 10 is positive. So decimal point will move 6 places on right. $1.24\times {10}^{6}=1240000$

4. Convert $0.00000067$in scientific notationAs the number is very small the exponent will be negative. $0.00000067=6.7\times {10}^{-7}$

5. Convert $56000000$in scientific notationAs the number is big the exponent will be positive. $56000000=5.6\times {10}^{7}$

## Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

- Bachelors in Mathematics, Physics, chemistry
- Masters in Mathematics, Physics, chemistry
- All branches of engineering and Medical.

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