Theory:

In the previous class, we have learned about the laws of exponent.
 
Let's have a quick recall of the laws of exponent.
 
The laws are:
1. Product law
According to the product law, the exponents can be added when multiplying two powers with the same base.
 
an×am=an+m, where \(a ≠ 0\) and \(a\), \(m\), \(n\) are integers.
Example:
\( 10 ^2 × 10 ^5\)
 
Here, the base \(10\) is same for powers. So we can add the exponents using the product law.
 
102×105=102+5=107
2. Quotient law
The quotient law states that we can divide two powers with the same base by subtracting the exponents.
 
an:am=anam=anm,n>m,a0;, where, \(a\), \(m\), \(n\) are integers.
Example:
415413=41513=42
 
108105=1085=103
3. Power law
The power law states that when a number is raised to a power of another power, we need to multiply the powers or exponents.
 
anm=an×m, where \(a ≠ 0\) and \(a\), \(m\), \(n\) are integers.
Example:
453=45×3=4151042=104+2=106
Powers with Negative Exponent
A number with negative exponent is equal to the reciprocal of the number with positive exponent.
 
That is, an=1an, here \(n\) is an integer.
  • If the negative number \((-1)\) raised to the negative odd power \((\)\(-1\)\(^\text{odd power}\)\()\), then the resultant value is negative \((-1)\).
  • If the negative number \((-1)\) raised to the negative even power \((\)\(-1\)\(^\text{even power}\)\()\), then the resultant value is positive \((1)\).
Example:
52=152=15×5=125
 
103=1103=110×10×10=11000