5.4: Zero Exponent Rule

In section 5.3, the exponent of the number in the numerator was always greater than the exponent of the number in the denominator.

In section 5.4, the exponent of the number in the numerator will be equal to the exponent of the number in the denominator.

Definition: The Zero Exponent Rule

For any real number \(a\), The Zero Exponent Rule is the following

\(a^0= 1\)

Idea:

From previous sections:

\[x^5 = x \cdot x \cdot x \cdot x \cdot x \nonumber \]

and

\[\dfrac{x^5 }{x^5} =\dfrac{ x \cdot x \cdot x \cdot x \cdot x }{x \cdot x \cdot x \cdot x \cdot x }= \dfrac{\cancel{x \cdot x \cdot x \cdot x \cdot x }}{\cancel{x \cdot x \cdot x \cdot x \cdot x }}= 1 \nonumber \]

Hence,

\[\dfrac{x ^5 }{x^5} = x^{5−5 }= x^0=1 \nonumber \]

Example 5.4.1

Use the zero exponent rule to simplify expressions.

1. \(\dfrac{x^9 }{x^9}\)
2. \(\dfrac{d^5 }{d^2 \cdot d^3}\)
3. \(\dfrac{5(xy)^3 }{(xy)^3}\)
4. \(-\dfrac{y^3 }{\sqrt{5}y^3}\)
5. \(\dfrac{(ab^2 )^7 }{(ab^2)^2 \cdot (ab^2)^4 ˙(ab^2)}\)

Solution

Expression	Zero Exponent Rule
\(\dfrac{x^9 }{x^9}\)	\(x^{9−9} = x^0 = 1\)
\(\dfrac{d^5 }{d^2 \cdot d^3}\)	\(\dfrac{d^5 }{d^{2+3 }}= \dfrac{d^5 }{d^5} = d^{5−5 }= d^{0} = 1\)
\(\dfrac{5(xy)^3 }{(xy)^3}\)	\(5 \cdot \dfrac{(xy)^3 }{(xy)^3 }= 5 \cdot (xy)^{3−3 }= 5 \cdot (xy)^0 = 5 \cdot 1 = 5 \) The constant 5, can be factored out to see common bases clearly.
\(-\dfrac{y^3 }{\sqrt{5}y^3}\)	\(− \dfrac{1}{ \sqrt{5}} \cdot \dfrac{y^3 }{y^3 }= − \dfrac{1}{ \sqrt{5}} \cdot y ^{3−3 }= − \dfrac{1}{ \sqrt{5}} \cdot y^0 = − \dfrac{1}{ \sqrt{5}} \cdot 1 = − \dfrac{1}{ \sqrt{5}}\) The constant \(−\left( \dfrac{1 }{\sqrt{5}}\right )\), can be factored out to see common bases clearly.
\(\dfrac{(ab^2 )^7 }{(ab^2)^2 \cdot (ab^2)^4 ˙(ab^2)}\)	\(\dfrac{(ab^2 )^7 }{(ab^2)^{2+4+1}}= \dfrac{(ab^2 )^7}{ (ab^2)^7} = (ab^2 )^{7−7 }= (ab^2 )^0 = 1\) First, simplify the denominator using the product rule of exponents. Then use the quotient rule of exponents to simplify the remaining expression.

Note: \(0^0\) does not equal 1. This is a special case that is covered in advanced courses. For now consider \(0^0\) to be undefined.

Helpful steps to simplify expressions with exponents

1. Identify common bases.
2. If needed combine common bases using the product rule of exponents.
3. If the expression contains common bases in both the numerator and denominator, use the quotient rule of exponents as needed.

Exercise 5.4.1

Use all the rules of exponents covered so far in this chapter to simplify the following.

1. \(\dfrac{z ^4 }{z^ 4}\)
2. \(\dfrac{d^2 \cdot d^8}{ d^7 \cdot d^3}\)
3. \(\dfrac{5(x + y)^3 }{2(x + y)^3}\)
4. \(−\dfrac{\sqrt{9}{y^3 }}{y^3}\)
5. \(\dfrac{(a^3b^2 )^9}{ (a^3b^2)^3 \cdot (a^3b^2)^4 ˙(a^3b^2)^2}\)
6. \(\dfrac{(xyz)^{19} }{(xyz)^{19}}\)