To solve the equation [tex]\((-20)^{16} \cdot (-20)^x = (-20)^2\)[/tex], we will use the properties of exponents.
Step-by-Step Solution:
1. Identify the property of exponents to use:
When multiplying exponential expressions with the same base, [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]. Here, the base is [tex]\(-20\)[/tex].
2. Rewrite the equation using the property:
[tex]\[
(-20)^{16} \cdot (-20)^x = (-20)^{16+x}
\][/tex]
3. Equate the exponents:
Once we have the expressions on both sides with the same base, we can set the exponents equal to each other:
[tex]\[
16 + x = 2
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Subtract 16 from both sides of the equation to isolate [tex]\(x\)[/tex]:
[tex]\[
x = 2 - 16
\][/tex]
Simplify the result:
[tex]\[
x = -14
\][/tex]
So, the value of [tex]\(x\)[/tex] is [tex]\(-14\)[/tex].
