Exponents and Powers: The width of our galaxy, the Milky way from edge to edge, is \(946,000,000,000,000,000\,\,{\rm{km}}\). Scientists who studied lunar samples from Apollo space missions estimate that the moon is \({\rm{4,600,}}\,{\rm{000,}}\,{\rm{000}}\) years old. Can you read both of these numbers? To express these large numbers in a form, which can be read, written, compared easily, we use the exponential form.

When a number is multiplied with itself several times, it can be written in short form as under: \({\rm{2}} \times {\rm{2}} \times {\rm{2}} \times {\rm{2}} \times {\rm{2 = }}{{\rm{2}}^5}\). This short-form \({{\rm{2}}^5}\) is called an exponential expression. The factor multiplied by itself, again and again, is the base, and the number of times the factor appears is the exponent. In \({{\rm{2}}^5},\,2\) the base and \(5\) the exponent. We read \({{\rm{2}}^5}\) as \(”\,2\) to the power \(5”\). In this article, we will learn about powers and exponents.

Exponents and Powers

As we know that the continued sum of a number added to itself several times can be written as the product of the numbers, equal to the number of times it is added and the number itself.
For example,
\(2\, + \,2\, + \,2\, + 2\, + \,2 + 2 + 2 = 7 \times 2\)

Similarly, the continued product of a number multiplied with itself several times can be written as the number raised to the power a natural number, equal to the number of times the number is multiplied with itself.
For example, \(3 \times 3 \times 3 \times 3\) can be written as \({3^4}\) and it is read as \(”\,3\) raised to the power \(”\,4\) or fourth power of \(3\). In \({3^4}\) we call \(3\) as the base and \(4\) as the exponent or power.

In general, for any rational number \(x\) and a positive integer \(n\), we define
\({x^n} = x \times x \times x \times …. \times x\) (\(n\) times)
\({x^n}\) is called the \({n^{th}}\) power of \(x\) and is also read as \(x\) raised to the power \(n\). The rational number \(‘x’\) is called the base, and \(n\) is called the exponent or power or index.

Examples of Exponents and Powers

We have learnt that if \(m\) is a positive integer and a non-zero number \(b\) is multiplied \(m\) times, then we get \(m\) power of \(b\), which is denoted by \({b^m}\) is called \(m\) power of \(b\) or \(b\) raised to power \(m\).

For example,
Mass of earth is \(5,970,000,000,000,000,000,000,000\,\,{\rm{kg}}\) Large numbers more conveniently using exponents, as, \(5.97\, \times {10^{24}}\,{\rm{kg}}\)
We read \({10^{24}}\) as \(10\) raised to the power \(24\).
We know \({{\rm{7}}^4}\, = \,7 \times 7 \times 7 \times 7\)
And \({3^n}\, = \,3 \times 3 \times 3 \times 3 \times \,…\, \times 3 \times 3\,…(n\,\,times)\)

Note: Power \(2\) and power \(3\) have special names assigned to them.
POWER \(2\) is called SQUARE.
POWER \(3\) is called CUBE.

Function of Exponents and Powers

In exponential functions, a fixed base is raised to a variable exponent. In power functions, however, a variable base is raised to a fixed exponent. The parameter \(b\), called either the exponent or the power, determines the function’s rates of growth or decay.

Here, the graph of function \(y = {2^x}\) shows exponential growth for the various value of \(x\) and the graph of function \(y = {x^{ – 2}}\) shows exponential decay for the various value of \(x\).

Exponents and Powers Rules

Rules of exponents and powers show how to solve different types of math equations and how to add, subtract, multiply and divide exponents.

Rule 1: Multiplication of powers with a common base

The law implies that if the exponents with the same bases are multiplied, then exponents are added together.
The general form of this law is \({a^m} \times {a^n} = {a^{m + n}}.\)

Rule 2: Dividing exponents with the same base

In the division of exponential numbers with the same base, we need to do subtraction of exponents.
The general form of this law is \({a^m}\, \div \,\,{a^n}\, = \,{a^{m\, – \,n}}\,\)

Rule 3: The law of the power of a power

This law implies that we need to multiply the powers in case an exponential number is raised to another power.
The general form of this law is \({({a^m})^n}\, = \,{a^{m\, \times \,n}}\).

Rule 4: The law of multiplication of powers with different bases but same exponents.

This law states that we need to multiply the different bases and raise the same exponent to the product of bases.
The general form of this law is \({a^m}\, \times \,\,{b^m}\, = \,{(a\, \times \,b)^m}\).

Rule 5: The law of the division of powers with different bases but same exponents.

This law states that we need to divide the different bases and raise the same exponent to the quotient of bases.
The general form of this law is \({a^m}\, \div \,{b^m}\, = \,\frac{{{a^m}}}{{{b^m}}} = {\left( {\frac{a}{b}} \right)^m}\).

Rule 6: The law of negative exponents.

When an exponent is negative, we change it to positive by taking the reciprocal of an exponential number.
The general form of this law is \({a^{ – n}} = \,\frac{1}{{{a^n}}}\).

Rule 7: Zero power rule

Any base raised to the power of zero is equal to one.
The general form of this law is \({a^0}\, = \,1\).

Negative Power of a Non-zero Rational Number

Working Rule: Use the following results, whichever is required.

1. If \(x\) is any integer and \(m\) is a whole number, then \({x^m}\, = \,x\, \times \,x \times x \times x \times \,…\,m\,\,times\) \({x^{ – m}}\, = \,\frac{1}{{{x^m}}} = \frac{1}{{x\, \times \,x \times x \times x \times \,…\,m\,\,times}}\)
2. If \(a\) and \(b\) are any two integers and \(m\) is a whole number, then \({\left( {\frac{a}{b}} \right)^m}\, = \frac{{{a^m}}}{{{b^m}}} = \frac{{a \times a \times a \times \,…\,m\,times}}{{b \times b \times b \times \,…\,m\,times}}\), where \(b\,\, \ne \,\,0\)
\({\left( {\frac{a}{b}} \right)^{ – m}}\, = \frac{{{b^m}}}{{{a^m}}} = \frac{{b \times b \times b \times \,…\,m\,times}}{{a \times a \times a \times \,…\,m\,times}}\), where \(a\,\, \ne \,\,0\)
3. If \(x\) is a non-zero number, then \({x^0}\, = \,1\)
4. If \(m\) and \(n\) are any two integers, then \(\frac{{{x^m}}}{{{x^n}}} = \,{x^{m – n}},\,x \ne \,\,0\)

Numbers in Standard Form

A number that is written as \(\left( {x \times {{10}^m}} \right)\)  is said to be in standard form if \(x\) is a decimal number such that \(1 \le x < 10\) and \(m\) is either a positive or a negative integer.
For example, express \(15360000000\) in standard form.

We have, \(15360000000\) can be expressed in standard form as \(1536 \times 10000000 = 1.536 \times 1000 \times {10^7}\)
\( = 1.536 \times {10^3} \times {10^7} = 1.536 \times {10^{10}}\)

Solved Examples

Q.1. Express \({\left( {\frac{3}{8}} \right)^{ – 2}} \times \,{\left( {\frac{4}{5}} \right)^{ – 3}}\) as a rational number of the form
Ans: we have, \({\left( {\frac{3}{8}} \right)^{ – 2}} \times \,{\left( {\frac{4}{5}} \right)^{ – 3}} = \frac{1}{{{{\left( {\frac{3}{8}} \right)}^2}}} \times \frac{1}{{{{\left( {\frac{4}{5}} \right)}^3}}}\,\,\left[ {\,{a^{ – n}} = \frac{1}{{{a^n}}}} \right]\)
\( = \frac{1}{{\frac{9}{{64}}}} \times \frac{1}{{\frac{{64}}{{125}}}}\)
\({\left( {\frac{3}{8}} \right)^{ – 2}} \times {\left( {\frac{4}{5}} \right)^{ – 3}} = \frac{{125}}{9}\)
Hence, the rational number is \(\frac{{125}}{9}.\)

Q.2. Simplify \({({5^{ – 1}} \times \,\,{3^{ – 1}})^{ – 1}} \div {6^{ – 1}}\).
Ans: Given,\({({5^{ – 1}} \times \,\,{3^{ – 1}})^{ – 1}} \div {6^{ – 1}}\)
\( = {\left( {\frac{1}{5} \times \frac{1}{3}} \right)^{ – 1}} \div \frac{1}{6}\,\,\,\,\,\,\,{(a)^{ – 1}} = \frac{1}{a}\)
\( = {\left( {\frac{1}{5} \times \frac{1}{3}} \right)^{ – 1}} \div \frac{1}{6}\,\,\)
\( = {\left( {\frac{1}{{15}}} \right)^{ – 1}} \div \frac{1}{6}\)
\( = \frac{1}{{\frac{1}{{15}}}} \div \frac{1}{6} = 15 \div \frac{1}{6}\)
\( = 15 \times \frac{6}{1} = 90\)
Hence, the simplified form is \(90.\)

Q.3.By what number should \({( – 6)^{ – 1}}\) be multiplied so that the product may be equal to \({( – 3)^{ – 1}}\)?
Ans: Let \({( – 6)^{ – 1}}\) be multiplied by \(x\) to get \({( – 6)^{ – 1}}\).
Then, \(x\,\, \times \,\,{( – 6)^{ – 1}} = \,{( – 3)^{ – 1}}\)
\( \Rightarrow x = {\left( { – 3} \right)^{ – 1}} \div {\left( { – 6} \right)^{ – 1}}\)
\( \Rightarrow x = \left( {\frac{1}{{ – 3}}} \right) \div \left( {\frac{{ – 6}}{1}} \right)\)
\( \Rightarrow x = \left( {\frac{1}{{ – 3}}} \right) \times \left( {\frac{{ – 6}}{1}} \right)\)
\( \Rightarrow x\, = 2\)
Hence, the required number is \(2.\)

Q.4. Evaluate \({\left( {\frac{3}{5}} \right)^{ – 3}} \times \,\,{\left( {\frac{3}{5}} \right)^5}\)
Ans: Given \( = \,{\left( {\frac{3}{5}} \right)^{ – 3}} \times \,\,{\left( {\frac{3}{5}} \right)^5}\)
\( = \,{\left( {\frac{3}{5}} \right)^{ – 3 + 5}} = \,\,{\left( {\frac{3}{5}} \right)^2}\)
\( = \frac{3}{5} \times \frac{3}{5} = \frac{9}{{25}}\)
\(\therefore {\left( {\frac{3}{5}} \right)^{ – 3}} \times {\left( {\frac{3}{5}} \right)^5} = \frac{9}{{25}}.\)

Q.5. By what number should \({\left( {\frac{{ – 1}}{2}} \right)^{ – 3}}\) be divided so that the quotient is \({\left( {\frac{{ – 3}}{4}} \right)^{ – 3}}\)
Ans: Let the required number be \(x\) Then
\( \Rightarrow \frac{{{{\left( {\frac{{ – 1}}{2}} \right)}^{ – 3}}}}{x} = {\left( {\frac{{ – 3}}{4}} \right)^{ – 2}}\)
\( \Rightarrow {\left( {\frac{2}{{ – 1}}} \right)^3} = {\left( {\frac{4}{{ – 3}}} \right)^2} \times x\)
\( \Rightarrow \frac{{{2^3}}}{{{{( – 1)}^3}}} = \frac{{{4^2}}}{{{{( – 3)}^2}}} \times x\)
\( \Rightarrow \frac{8}{{ – 1}} = \frac{{16}}{9} \times x\)
\( \Rightarrow x = \frac{9}{{ – 2}}\)
Hence, the required number is \(\frac{9}{{ – 2}}.\)

Summary

In this article, we learnt about the definition of exponents and powers, example of exponents and powers, function of exponents and powers, exponents and powers rules, solved examples on exponents and powers, frequently asked questions (FAQ) on exponents and powers.

The learning outcome of this article is how to write very large numbers in a convenient way to read and compare using exponents and powers.

FAQs on Exponents and Powers

Q.1. What are the \(7\) laws of exponents?
Ans: Law \(1\): Multiplication of powers with a common base
The general form of this law is \({a^m} \times {a^n}\, = {a^{m + n}}\)
Law \(2\): Dividing exponents with the same base
The general form of this law is \({a^m} \div {a^n}\, = {a^{m\, – \,n}}\)
Law \(3\): The law of the power of a power
The general form of this law is \({({a^m})^n}\, = {a^{m\, \times \,n}}\)
Law \(4\): The law of multiplication of powers with different bases but same exponents.
The general form of this law is \({a^m}\, \times {b^m}\, = {(a \times b)^m}\)
Law \(5\): The law of the division of powers with different bases but the same exponents.
The general form of this law is \({a^m}\, \div {b^m}\, = \frac{{{a^m}}}{{{b^m}}} = {\left( {\frac{a}{b}} \right)^m}\)
Law \(6\): The law of negative exponents.
The general form of this law is \({a^{ – n}}\, = \,\frac{1}{{{a^n}}}\)
Law \(7\): Zero power rule
The general form of this law is \(^0=1.\)

Q.2. What are exponent and power? Explain with example.
Ans: If \(m\) is a positive integer and a non-zero number \(b\) is multiplied \(m\) times, then we get \(m\)th power of \(b\) which is denoted by \({b^m}\). Here \(b\) is called the base, \(m\) is called the exponent or index, and \({b^m}\) is called \(m\)th power of \(b\) or \(b\) raised to power \(m\).
For example,
The distance of the moon from the Earth is \(384467000\,{\rm{m}}\)
By using exponents, we can write \(384467000\,{\rm{m}}\) as \(384467 \times {10^3}\,{\rm{m}}\)
\( = 3.84467 \times {10^5} \times {10^3}\,{\rm{m}}\)
\( = 3.84467 \times {10^8}\)
Hence, the exponential form of \( = 384467000\,\,{\rm{m}}\) is \( = 3.84467 \times {10^8}.\)
We read \({10^8}\) as \(10\) raised to the power \(8.\)

Q.3. What are the \(5\) laws of exponent?
Ans: The five laws of exponents are:
1. Multiplying Powers with the same base.
2. Dividing Powers with the same base.
3. Law of power of a power.
4. Multiplying Powers with the same Exponents.
5. Negative Exponents.

Q.4. What are the formulas of exponents and powers?
Ans: Formulas of exponents and powers
1. \({a^m} \times \,{a^n}\, = \,{a^{m + n}}\)
2. \({a^m}\, \div \,{a^n}\, = \,{a^{m – n}}\)
3. \({({a^m})^n} = {a^{m \times n}}\)
4. \({a^m}\, \times {b^m}\, = {(a \times b)^m}\)
5. \({a^m} \div {b^m} = \frac{{{a^m}}}{{{b^m}}} = {\left( {\frac{a}{b}} \right)^m}\)
6. \({a^{ – n}}\, = \,\frac{1}{{{a^n}}}\)
7. \({a^0} = 1\)
8. \({a^{\frac{1}{n}\,}} = \sqrt[n]{a}\)

Q.5. How do you calculate exponential powers?
Ans: The exponent of a number says how many times to use the number in a multiplication.
For example: \({8^3} = 8 \times 8 \times 8 = 512.\)

We hope this detailed article on powers and exponents helped you in your studies. If you have any doubts or queries on this topic, you can comment down below and we will be more than happy to help you.