Exponents And Powers

Exponents And Powers

Power with exponent zero law :
    The power with an exponent zero law of exponents states that any non-zero number raised to the power of zero is equal to 1. Mathematically, it can be expressed as:    
    \( a^0 = 1\left( {a \ne 0} \right) \)
    Here, a is a non-zero base, and the exponent is zero. The rule tells us that any non-zero number raised to the power of zero results in 1.
    For example:
     5⁰=1   
    \( \left( {\frac{2} {3}} \right)^0 = 1 \)
NOTE:     It's important to note that this rule is only defined for non-zero bases. The expression 0⁰ is often considered undefined or indeterminate in many contexts.
viii.  If \( x^m = x^n \), then m = n
    Ex : \( x^{6 + a} = x^5 \)
     \( \Rightarrow 6 + a = 5 \)   
     \( \Rightarrow a = 5 - 6 \)  
     \( \Rightarrow a = - 1 \)   
ix.  If \( a^m = b^m \) then a = b
    Ex : \( y^{x + 6} = 5^{x + 6} \Rightarrow y = 5 \)
Note : \( y^{x + 6} = 5^{x + 6} \) 
\( \Rightarrow \frac{{y^{x + 6} }} {{5^{x + 6} }} = 1 \Rightarrow \left( {\frac{y} {5}} \right)^{x + 6} = \left( {\frac{y} {5}} \right)^0 \)
\( \Rightarrow x + 6 = 0 \)
\( \Rightarrow x = - 6 \)