To simplify the expression [tex]\(\frac{4^9}{4^3}\)[/tex], follow these steps:
1. Understand the properties of exponents: One of the key properties of exponents is that when you divide two expressions with the same base, you subtract the exponents:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
2. Apply this property to the given expression: Here, the base is 4, the exponent in the numerator is 9, and the exponent in the denominator is 3. Applying the property mentioned, we get:
[tex]\[
\frac{4^9}{4^3} = 4^{9-3}
\][/tex]
3. Simplify the exponent: Subtract the exponent in the denominator (3) from the exponent in the numerator (9):
[tex]\[
9 - 3 = 6
\][/tex]
So, we can rewrite the expression as:
[tex]\[
4^6
\][/tex]
4. Evaluate the simplified expression: Now, calculate [tex]\(4^6\)[/tex]:
[tex]\[
4^6 = 4096
\][/tex]
Thus, the simplified result of the given expression [tex]\(\frac{4^9}{4^3}\)[/tex] is [tex]\(4^6 = 4096\)[/tex].
