Introduction to Exponents

We use the word exponentially to describe something that sees a very drastic change. Exponents mean that the number is multiplied by itself multiple times. x raised to y means that x is multiplied by itself y times. In this chapter, we will study about exponents and their laws.

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Exponents

We know that, \( 10+10+10+10+10=50 \) ………..(1)

whereas, \(10\times 10\times 10\times 10\times 10=100000 \) …….(2)

Equation 1 can also be written as – \(10+10+10+10+10=5\times 10 \) and read as 5 times 10.  Whereas equation 2 is also written as \(10\times 10\times 10\times 10\times 10=10^{5} \) and read as 10 raised to the power 5. In 10⁵, (10) is called the base and (5) the exponent (or index or power). 10⁵, is called the exponential form or power notation of 100000.

Similarly, 11+11+11 =3×11 read as three times 11 and \(11\times 11\times 11\) =11³ =1331 and is read as 11 raised to the power 3. In 11³, 11 is called the base and 3, the exponent. In general, if x is an integer, then n times the product of x with itself is given as (x) x(x) x(x)….n times and is denoted as xⁿ. Here x is called the base and n, the exponent or the power or the index.

\( X^{ n }=product\: of\: n\: factors\: each\: of\: which\: is\: X \)

For example, \(7^{ 6 }=7\times 7\times 7\times 7\times 7\times 7 \)

Power of 1

Remember the following two simple rules of 1.

• Any number raised to the power of ‘one’ equals itself. For example,\(15^{ 1 }=15 \)
• One raised to any power is one. This is so because one multiplied by one as many times is always equal to one. For example, \(1^{5}=1\times 1\times 1\times 1\times 1=1 \)

Power of -1

\( (-1)^{1}= -1\) and \( (-1)^{2}= (-1) \times (-1) =1 \)

In general,  \((-1) ^{odd\: number}=-1\) and \( (-1) ^{even\: number}=1 \)

Laws or Rules of Exponents

First Rule: Multiplying power with the same base

Example: \( (3)^{3}\times (3)^{4}=(3\times 3\times 3\times 3)\times (3\times 3\times 3)=(3\times 3\times 3\times 3\times 3\times 3\times 3)=3^{ 7 } \)

Now, carefully study the power notation in final steps of the above example. You will see that the index on the RHS is the sum of the indices on the LHS. Thus, if x is any integer and m and n are whole numbers, then \({ X }^{ m }\times X^{ n }=X^{ m+n } \).

Second Rule: Dividing power with the same base

Example: \(\frac { { 3 }^{ 4 } }{ { 3 }^{ 3 } } =\frac { { 3 }\times { 3 }\times { 3 }\times { 3 } }{ 3\times { 3 }\times { 3 } } =3 \)

Now study the power notation obtained from the example. You will find that the index on the RHS is the difference of indices on the LHS. In general, for any nonzero integer x, \(\frac { { X }^{ m } }{ { X }^{ n } } ={ X }^{ m-n } \), where m and n are whole numbers and m> n.

Third Rule: Power of a power

Example: \({ {( { 2 }^{ 2 }) } }^{ 3 }={ {( { 2 }^{ 2 }) } }\times { {( { 2 }^{ 2 }) \times } }{ {( { 2 }^{ 2 }) ={ 2 }^{ 2+2+2 }={ 2 }^{ 6 } } } \)

Study the pattern of power notations in the above examples and complete the general relation, \({( { X }^{ m }) }^{ n }={ X }^{ n\times m } \), where x is any nonzero integer and m,n are whole numbers. Thus, to raise a power to power, multiply the exponents.

Fourth Rule: Numbers with exponent zero

Example:\(\frac { { 3 }^{ 4 } }{ { 3 }^{ 4 } } =\frac { { 3 }\times { 3 }\times { 3 }\times { 3 } }{ 3\times { 3 }\times { 3 }\times { 3 } } =1 \)

The result can also be determined using second rule for exponents. i.e. \(\frac { { 3 }^{ 4 } }{ { 3 }^{ 4 } } ={ 3 }^{ 4-4 }={ 3 }^{ 0 }=1 \). Thus, any number (except 0) raised to the power 0 is 1, \(X^{ 0 }=1 \)

Fifth Rule: Multiplying bases with the same exponents

Example: \( (3\times 4)^{2}= (3\times 4) \times (3\times 4) =( 3\times 3) \times ( 4\times 4)=(3)^{2}\times (4)^{ 2 } \)

Thus, if x, y are any integers and m is whole number, then \({( x\times y) }^{ m }=x^{ m }\times { y }^{ m } \)

Sixth Rule: Dividing bases with the same exponents

Example: \(\frac { { 5 }^{ 2 } }{ { 3 }^{ 2 } } =\frac { 5\times 5 }{ 3\times 3 } =\frac { 5 }{ 3 } \times \frac { 5 }{ 3 } = ( \frac { 5 }{ 3 }) ^{ 2 } \)

Thus, if x and y are integers and m is the whole number, then \(\frac { X^{ m } }{ Y^{ m } } ={( \frac { X }{ Y }) }^{ m } \)

Negative Exponent

In the sequence x^5, x^4,x³ ……, each term is obtained by subtracting 1 from the exponents of its preceding term. Then, the next five terms of the sequence will be x, x_­­­⁰,x^-1,x^-2,x^-3 and so on.

An alternative way to determine successive terms in the sequence is by dividing the preceding term by x. thus, in the given sequence, the next five terms following x² will be x, 1, 1/x, 1/x², 1/x³. You already know that x⁰= 1. From the above explanation, you can conclude that-

\(X^{ -1 }=\frac { 1 }{ X } \\ X^{ -m }=\frac { 1 }{ { X }^{ m } } \)

Solved Examples for You

Question 1: Find the value of \( \frac { { 2 }^{ 8 }\times { 9 }^{ 3 } }{ { 4 }^{ 3 }\times { 3 }^{ 7 } }  \)

Answer : By using rule 2 and 3 –

\(x=\frac { { 2 }^{ 8 }\times { 9 }^{ 3 } }{ { 4 }^{ 3 }\times { 3 }^{ 7 } } \\ x=\frac { { 2 }^{ 8 }\times {( 3\times 3) }^{ 3 } }{ { ( 2\times 2) }^{ 3 }\times { 3 }^{ 7 } } \\ x=\frac { { 2 }^{ 8 }\times {( 3^{ 2 }) }^{ 3 } }{ {( 2^{2}) }^{ 3 }\times { 3 }^{ 7 } } \\ x=\frac { { 2 }^{ 8 }\times { 3 }^{ 6 } }{ {( 2^{6}) }\times { 3 }^{ 7 } } \\ x={ 2 }^{ 8-6 }\times { 3 }^{ 6-7 }\\ x={ 2 }^{ 2 }\times { 3 }^{ -1 }\\ x=\frac { 4 }{ 3 } \)

Question 2: How to solve the exponents with exponents?

Answer: Firstly, we have to raise the expressions in the parentheses to their powers. After that, we have to multiply the 2 expressions together. We get to see multiplying the exponents and adding exponents.

Question 3: How to multiply the exponents?

Answer: Multiplying the exponents with multiple bases:

First of all, multiply all the bases together. Secondly, add on the exponent and instead of adding the 2 exponents together keep that equivalent. This happens because of the 4th exponent rule that says ‘distribute the power to every single base while raising numerous variables by a power’.

Question 4: What are the 7 different laws of exponents?

Answer: The 7 different laws of exponents are as follows:

1. Multiplying the powers with equal base.
2. Dividing the powers with a similar base.
3. Power of a power.
4. Multiplying the powers with similar exponents.
5. Negative (-) Exponents.
6. Power with the exponent zero (0).