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To determine which expressions are equivalent to [tex]\( 9^x \)[/tex], we need to simplify each expression and compare it to [tex]\( 9^x \)[/tex].
### Expression A: [tex]\( 9 \cdot 9^{x+1} \)[/tex]
We can use the properties of exponents to simplify this expression:
[tex]\[ 9 \cdot 9^{x+1} = 9^1 \cdot 9^{x+1} = 9^{1} \cdot 9^{x} \cdot 9^{1} = 9^{x+2} \][/tex]
Clearly, [tex]\( 9 \cdot 9^{x+1} = 9^{x+2} \neq 9^x \)[/tex]. Therefore, Expression A is not equivalent.
### Expression B: [tex]\( 9 \cdot 9^{x-1} \)[/tex]
Using the properties of exponents again:
[tex]\[ 9 \cdot 9^{x-1} = 9^1 \cdot 9^{x-1} = 9^{1+(x-1)} = 9^{x} \][/tex]
Expression B simplifies directly to [tex]\( 9^x \)[/tex]. Therefore, Expression B is equivalent.
### Expression C: [tex]\( x^5 \)[/tex]
This expression is already simplified, and it takes the form [tex]\( x^5 \)[/tex], which is very different from [tex]\( 9^x \)[/tex]. Therefore, Expression C is not equivalent.
### Expression D: [tex]\( \left(\frac{36}{4}\right)^x \)[/tex]
We can first simplify the base inside the parentheses:
[tex]\[ \frac{36}{4} = 9 \][/tex]
Thus:
[tex]\[ \left(\frac{36}{4}\right)^x = 9^x \][/tex]
Expression D simplifies directly to [tex]\( 9^x \)[/tex]. Therefore, Expression D is equivalent.
### Expression E: [tex]\( \frac{36^x}{4} \)[/tex]
This expression can be analyzed by separating the exponents, but it will not match [tex]\( 9^x \)[/tex]:
[tex]\[ \frac{36^x}{4} \neq 9^x \][/tex]
To see why this does not work, consider that:
[tex]\[ \frac{36}{4} = 9 \][/tex]
However, when exponentiated:
[tex]\[ \left(\frac{36}{4}\right)^x = 9^x \][/tex]
This is not the same form as dividing the exponential:
[tex]\[ \frac{36^x}{4} \neq 9^x \][/tex]
Therefore, Expression E is not equivalent.
### Expression F: [tex]\( \frac{36^x}{4^x} \)[/tex]
We can simplify this using the properties of exponents:
[tex]\[ \frac{36^x}{4^x} = \left(\frac{36}{4}\right)^x = 9^x \][/tex]
Expression F simplifies directly to [tex]\( 9^x \)[/tex]. Therefore, Expression F is equivalent.
### Summary
The expressions equivalent to [tex]\( 9^x \)[/tex] are:
- [tex]\( 9 \cdot 9^{x-1} \)[/tex] (Expression B)
- [tex]\( \left(\frac{36}{4}\right)^x \)[/tex] (Expression D)
- [tex]\( \frac{36^x}{4^x} \)[/tex] (Expression F)
Thus, the correct answers are B, D, and F.
### Expression A: [tex]\( 9 \cdot 9^{x+1} \)[/tex]
We can use the properties of exponents to simplify this expression:
[tex]\[ 9 \cdot 9^{x+1} = 9^1 \cdot 9^{x+1} = 9^{1} \cdot 9^{x} \cdot 9^{1} = 9^{x+2} \][/tex]
Clearly, [tex]\( 9 \cdot 9^{x+1} = 9^{x+2} \neq 9^x \)[/tex]. Therefore, Expression A is not equivalent.
### Expression B: [tex]\( 9 \cdot 9^{x-1} \)[/tex]
Using the properties of exponents again:
[tex]\[ 9 \cdot 9^{x-1} = 9^1 \cdot 9^{x-1} = 9^{1+(x-1)} = 9^{x} \][/tex]
Expression B simplifies directly to [tex]\( 9^x \)[/tex]. Therefore, Expression B is equivalent.
### Expression C: [tex]\( x^5 \)[/tex]
This expression is already simplified, and it takes the form [tex]\( x^5 \)[/tex], which is very different from [tex]\( 9^x \)[/tex]. Therefore, Expression C is not equivalent.
### Expression D: [tex]\( \left(\frac{36}{4}\right)^x \)[/tex]
We can first simplify the base inside the parentheses:
[tex]\[ \frac{36}{4} = 9 \][/tex]
Thus:
[tex]\[ \left(\frac{36}{4}\right)^x = 9^x \][/tex]
Expression D simplifies directly to [tex]\( 9^x \)[/tex]. Therefore, Expression D is equivalent.
### Expression E: [tex]\( \frac{36^x}{4} \)[/tex]
This expression can be analyzed by separating the exponents, but it will not match [tex]\( 9^x \)[/tex]:
[tex]\[ \frac{36^x}{4} \neq 9^x \][/tex]
To see why this does not work, consider that:
[tex]\[ \frac{36}{4} = 9 \][/tex]
However, when exponentiated:
[tex]\[ \left(\frac{36}{4}\right)^x = 9^x \][/tex]
This is not the same form as dividing the exponential:
[tex]\[ \frac{36^x}{4} \neq 9^x \][/tex]
Therefore, Expression E is not equivalent.
### Expression F: [tex]\( \frac{36^x}{4^x} \)[/tex]
We can simplify this using the properties of exponents:
[tex]\[ \frac{36^x}{4^x} = \left(\frac{36}{4}\right)^x = 9^x \][/tex]
Expression F simplifies directly to [tex]\( 9^x \)[/tex]. Therefore, Expression F is equivalent.
### Summary
The expressions equivalent to [tex]\( 9^x \)[/tex] are:
- [tex]\( 9 \cdot 9^{x-1} \)[/tex] (Expression B)
- [tex]\( \left(\frac{36}{4}\right)^x \)[/tex] (Expression D)
- [tex]\( \frac{36^x}{4^x} \)[/tex] (Expression F)
Thus, the correct answers are B, D, and F.
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