To determine which expressions are equivalent to [tex]\(10^x\)[/tex], let's analyze each option one by one.
### Option A: [tex]\(x^5\)[/tex]
This expression is clearly not equivalent to [tex]\(10^x\)[/tex], as [tex]\(x\)[/tex] raised to the power of 5 does not equate to 10 raised to the power of [tex]\(x\)[/tex].
### Option B: [tex]\(\frac{50 \times 5}{5}\)[/tex]
Simplifying this expression:
[tex]\[
\frac{50 \times 5}{5} = 50
\][/tex]
This does not equal [tex]\(10^x\)[/tex] for any [tex]\(x\)[/tex]. This is a constant value.
### Option C: [tex]\(10 \cdot 10^{x+1}\)[/tex]
Using the properties of exponents, we can rewrite this as:
[tex]\[
10 \cdot 10^{x+1} = 10^1 \cdot 10^{x+1} = 10^{1 + (x+1)} = 10^{x+1+1} = 10^{x+2}
\][/tex]
This is clearly not equivalent to [tex]\(10^x\)[/tex].
### Option D: [tex]\(\left(\frac{50}{5}\right)^x\)[/tex]
Simplifying inside the parentheses first:
[tex]\[
\left(\frac{50}{5}\right)^x = (10)^x = 10^x
\][/tex]
So this expression is indeed equivalent to [tex]\(10^x\)[/tex].
### Option E: [tex]\(10 \cdot 10^{x-1}\)[/tex]
Using the properties of exponents, we can rewrite this as:
[tex]\[
10 \cdot 10^{x-1} = 10^1 \cdot 10^{x-1} = 10^{1 + (x-1)} = 10^{x}
\][/tex]
This is indeed equivalent to [tex]\(10^x\)[/tex].
### Option F: [tex]\(\frac{50^x}{5^x}\)[/tex]
We can rewrite the fraction using the properties of exponents:
[tex]\[
\frac{50^x}{5^x} = \left(\frac{50}{5}\right)^x = (10)^x = 10^x
\][/tex]
So this expression is also equivalent to [tex]\(10^x\)[/tex].
### Conclusion
Expressions D, E, and F are equivalent to [tex]\(10^x\)[/tex].
Thus, the equivalent expressions are:
- D. [tex]\(\left(\frac{50}{5}\right)^x\)[/tex]
- E. [tex]\(10 \cdot 10^{x-1}\)[/tex]
- F. [tex]\(\frac{50^x}{5^x}\)[/tex]
