To find the value of [tex]\( x \)[/tex] in the equation [tex]\( 5^{x+2} = 5^9 \)[/tex], follow these steps:
1. Understand the properties of exponents: When the bases are the same, you can set the exponents equal to each other. This principle comes from the property of exponents that states if [tex]\(a^m = a^n\)[/tex], then [tex]\(m = n\)[/tex], provided that [tex]\(a\)[/tex] is a non-zero number.
2. Apply this property to the given equation:
[tex]\[
5^{x+2} = 5^9
\][/tex]
Since the bases ([tex]\(5\)[/tex]) are the same, we can equate the exponents:
[tex]\[
x + 2 = 9
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x + 2 = 9
\][/tex]
4. Subtract 2 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x = 9 - 2
\][/tex]
5. Calculate the result:
[tex]\[
x = 7
\][/tex]
Thus, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\( 5^{x+2} = 5^9 \)[/tex] is [tex]\( \boxed{7} \)[/tex].
