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Powers with Negative Exponents

An Introduction to Powers with Negative Exponents

Last updated date: 17th Mar 2023
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Large numbers like \[75,000,000\], \[1,459,500,000,000\], \[5,978,043,000,000,000\], etc. are difficult for us to read, understand, and compare. We take the help of exponents to make such large numbers simple to read, understand, and compare. For example, the number \[6 \times 6 \times 6 \times 6\] is understood as $6$ raised to the power $4$. In the number ${6^4}$, the exponent is $4$ and the base is $6$.


This notation is also known as power notation or exponential form. Since \[{10^2} = 10 \times 10 = 100\], \[{10^1} = 10 = \dfrac{{100}}{{10}}\], \[{10^0} = 1 = \dfrac{{10}}{{10}}\], and \[{10^{ - 1}} = \dfrac{1}{{10}}\], we know that. The base $10$’s negative exponent, $ - 1$, is used here. The value becomes one-tenth of the previous value when the exponent reduces by $1$. The powers with negative exponents, their characteristics, and the problems based on them will all be covered in this article.


What are Exponents?

Exponential form refers to the use of powers to reduce large numbers into a more simplified form.

An example is \[10000 = 10 \times 10 \times 10 \times 10 = {10^4}\] .

\[10 \times 10 \times 10 \times 10\] is represented by the abbreviation ${10^4}$ . In this case, $10$ serves as the base and $4$ as the exponent.


Rules of Negative Exponents

We have a series of principles or rules for negative exponents which make the process of simplification easy. The fundamental guidelines for resolving negative exponents are listed below.


Rule 1: The reciprocal of a base, which is $\dfrac{1}{a}$, is multiplied by itself $n$ times according to the negative exponent rule for bases with the negative exponent $ - n$.

Specifically, ${a^{( - n)}} = \dfrac{1}{a} \times \dfrac{1}{a} \times \ldots n$ times which equals $\dfrac{1}{{{a^n}}}$.

Rule 2: The rule holds true even when the denominator contains a negative exponent.

In other words, $\dfrac{1}{{{a^{( - n)}}}} = a \times a \times \ldots n$ which is equal to ${a^n}$ .


Negative Laws of Exponents

  • According to the first law, if $m$ and $n$ are natural numbers and $a$ is any non-zero rational integer, then ${a^m} \times {a^n} = {a^{m + n}}$ .

${3^{ - 3}} = \dfrac{1}{{{3^3}}}$ and ${3^{ - 2}} = \dfrac{1}{{{3^2}}}$ are known numbers.

In light of this, ${3^{ - 3}} \times {3^{ - 2}} = \dfrac{1}{{{3^3}}} \times \dfrac{1}{{{3^2}}} = \dfrac{1}{{{3^{3 + 2}}}} = \dfrac{1}{{{3^5}}} = {3^{ - 5}}$ .

Clearly, adding $ - 3$ and $ - 2$ results in $ - 5$ . Therefore, the first law, ${a^m} \times {a^n} = {a^{m + n}}$, still applies to negative exponents. Alternatively, ${a^{ - m}} \times {a^{ - n}} = {a^{ - (m + n)}}$ .

  • According to the second law, if $a$ is any non-zero rational number and $m$, $n$ are natural numbers with $m > n$, then $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$ or ${a^m} \div {a^n} = {a^{m - n}}$.

Now, think about ${2^{ - 3}}$ and ${2^{ - 2}}$.

${2^{ - 3}} \div {2^{ - 2}} = \dfrac{1}{{{2^3}}} \div \dfrac{1}{{{2^2}}} = \dfrac{1}{{{2^3}}} \times \dfrac{{{2^2}}}{1} = {2^{2 - 3}} = {2^{ - 1}}$ .

Therefore, it is evident that $(2 - 3) = - 1$. This suggests that the second law, ${a^m} \div {a^n} = {a^{m - n}}$, is valid for negative exponents. Alternatively, ${a^{ - m}} \div {a^{ - n}} = {a^{ - (m - n)}}$.

  • The third law asserts that ${({a^m})^n} = {a^{m \times n}} = {({a^n})^m}$ if $a$ is any non-zero rational integer and $m$, $n$ are natural numbers.

Hence, \[{({2^3})^2} = {2^{3 \times 2}} = {2^6}\] . So, \[{({2^{ - 3}})^{ - 2}} = {2^{( - 3) \times ( - 2)}} = {2^6}\].

As a result, the third law, ${({a^m})^n} = {a^{m \times n}} = {({a^n})^m}$, is valid for negative exponents. The formula is ${({a^{ - m}})^{ - n}} = {a^{( - m) \times ( - n)}} = {({a^{ - n}})^{ - m}}$.

  • According to the fourth law, if $a$ and $b$ are rational non-zero numbers and $n$ is a natural number, then ${a^n} \times {b^n} = {(ab)^n}$ .

Consider the following:

${2^{ - 3}} \times {3^{ - 3}} = \dfrac{1}{{{2^3}}} \times \dfrac{1}{{{3^3}}} = \dfrac{1}{{{{(2 \times 3)}^3}}} = {(2 \times 3)^{ - 3}}$

Because of this, the fourth law, ${a^n} \times {b^n} = {(ab)^n}$, still applies to negative exponents. It can be written as ${a^{ - n}} \times {b^{ - n}} = {(ab)^{ - n}}$.

  • If $a$ and $b$ are non-zero rational numbers and $n$ is a natural number, the fifth law states that $\dfrac{{{a^n}}}{{{b^n}}} = {\left( {\dfrac{a}{b}} \right)^n}$.

Consider the expression $\dfrac{{{4^{ - 3}}}}{{{5^{ - 3}}}} = \dfrac{{{5^3}}}{{{4^3}}} = \dfrac{{5 \times 5 \times 5}}{{4 \times 4 \times 4}} = {\left( {\dfrac{5}{4}} \right)^3} = {\left( {\dfrac{4}{5}} \right)^{ - 3}}$ .

As a result, $\dfrac{{{a^n}}}{{{b^n}}} = {\left( {\dfrac{a}{b}} \right)^n}$, the fifth law still applies to negative exponents. Hence, $\dfrac{{{a^{ - n}}}}{{{b^{ - n}}}} = {\left( {\dfrac{a}{b}} \right)^{ - n}}$ .


How to Solve Negative Exponents

Use one of the following rules to change negative exponents into positive exponents before simplifying equations with negative exponents:

  • \[{a^{ - n}} = \dfrac{1}{{{a^n}}}\]

  • \[\dfrac{1}{{{a^{ - n}}}} = {a^n}\]


Negative Exponents Examples

1. Find the solution to the expression ${({3^2} + {4^2})^{ - 2}}$.

Solution: The provided expression is

\[{\left( {{3^2} + {4^2}} \right)^{ - 2}} = {\left( {9 + 16} \right)^{ - 2}}\]

\[ = {\left( {25} \right)^{ - 2}}\]

\[ = \dfrac{1}{{{{25}^2}}}\] (by negative exponents rule)

$ = \dfrac{1}{{625}}$

Because of this, ${({3^2} + {4^2})^{ - 2}} = \dfrac{1}{{625}}$.


2. Condense ${({2^{ - 1}} \div {5^1})^2} \times {\left( {\dfrac{{ - 5}}{8}} \right)^{ - 1}}$ .

Solution: The number is ${({2^{ - 1}} \div {5^1})^2} \times {\left( {\dfrac{{ - 5}}{8}} \right)^{ - 1}}$

because ${a^{ - 1}} = \dfrac{1}{a}$ $ \Rightarrow {\left( {\dfrac{1}{2} \div \dfrac{1}{5}} \right)^2} \times \dfrac{1}{{\dfrac{{ - 5}}{8}}}$

$ = {\left( {\dfrac{1}{2} \times \dfrac{5}{1}} \right)^2} \times \dfrac{8}{{ - 5}}$

$ = {\left( {\dfrac{5}{2}} \right)^2} \times \dfrac{8}{{ - 5}}$

Because $\dfrac{{{a^n}}}{{{b^n}}} = {\left( {\dfrac{a}{b}} \right)^n}$

$ = \dfrac{5}{4} \times \dfrac{8}{{ - 1}}$

$ = \dfrac{5}{1} \times \dfrac{2}{{ - 1}}$

\[ = - 10\]

Consequently, ${({2^{ - 1}} \div {5^1})^2} \times {\left( {\dfrac{{ - 5}}{8}} \right)^{ - 1}} = - 10$.

Practice Questions

  • Using the principles for negative exponents, find the unknown $x$ in the equation ${5^x} = \dfrac{1}{{625}}$

  • When \[\dfrac{{27}}{{{3^{ - x}}}} = {3^6}\] , what is the value of $x$?


Answers

  • $ - 4$

  • $x = 3$


Summary

The definition of exponents, various negative exponent laws, and power with exponent negative were all covered in the article above. Additionally, we have mastered the definition of power with negative exponents, and the power laws with negative exponents, and have worked through a few practice problems on negative exponents.

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FAQs on Powers with Negative Exponents

1. Why do negative exponents reciprocate?

The reciprocal of the provided integer is what we are expected to use when we need to convert a negative exponent to a positive one. Therefore, the opposite of the provided number is implied by the negative sign on an exponent, just as the positive sign on an exponent denotes the base's repeated multiplication. It can be expressed as the positive reciprocal of each negative exponent. A reciprocal is a fraction with a numerator and denominator that are inverted. The base's positive reciprocal times x can be used to represent negative exponents.

2. How can we tell if the exponent in scientific notation is negative?

When the exponent is positive, the decimal point is moved that many places to the right. When the exponent is negative, the decimal point is moved left by that many places. The digit word in scientific notation shows how many significant figures there are in the given number. For the scientific notation, the power is negative if the decimal number is small (less than 1, in absolute value), while the exponent is positive if the decimal number is high (greater than 1, in absolute value).

3. Where do exponents come into play in daily life?

Exponents are utilised in many different fields, including pH and Richter Measuring Scales, Computer Game Physics, Science, Engineering, Economics, Accounting, and Finance. Exponential growth is essential to a variety of areas, including finances, demography, biology, economics, resources, technology, and many more. Exponents are necessary for engineers because they frequently work with large and small numbers. Tensile strength, which quantifies the amount of stress that a structure can withstand, is calculated by engineers using exponents. Exponents are frequently used in daily life by medical professionals such as doctors and scientists who research medicine to define precise sums, computations, and words.