Certainly! Let's break down and solve the given problem step-by-step.
We are given the expression:
[tex]\[
\frac{5^9}{5^3}
\][/tex]
and we need to simplify this expression to find the value of [tex]\( a \)[/tex] in the equation:
[tex]\[
5^a
\][/tex]
### Step-by-Step Solution:
1. Understanding the Properties of Exponents:
The expression can be simplified using the properties of exponents. One useful property is:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
This property states that when you divide two exponents with the same base, you subtract the exponent in the denominator from the exponent in the numerator.
2. Applying the Property:
Here, the base is 5, and we have the exponents 9 and 3. Applying the property:
[tex]\[
\frac{5^9}{5^3} = 5^{9-3}
\][/tex]
3. Performing the Subtraction:
Calculate the exponent:
[tex]\[
9 - 3 = 6
\][/tex]
4. Writing the Simplified Expression:
The simplified expression for [tex]\(\frac{5^9}{5^3}\)[/tex] is:
[tex]\[
5^6
\][/tex]
5. Identifying [tex]\( a \)[/tex]:
Comparing this with [tex]\(5^a\)[/tex]:
[tex]\[
5^a = 5^6
\][/tex]
Therefore, [tex]\( a \)[/tex] is:
[tex]\[
a = 6
\][/tex]
### Conclusion:
The value of [tex]\( a \)[/tex] is [tex]\( 6 \)[/tex]. So, we can complete the expression as:
[tex]\[
a = 6
\][/tex]
