Respuesta :
Answer:
A- 8*8^(x-1)
C-(32/4)^x
F-32^x/4^x
Step-by-step explanation:
The expressions out of the specified expression which are equivalent to 8^x are given by:
- B. 32^{x/4}
- E. 32^x/4^x
- F. 8 * 8^{x-1}
What are equivalent expressions?
Those expressions who might look different but their simplified forms are same expressions are called equivalent expressions.
To derive equivalent expressions of some expression, we can either make it look more complex or simple.
What are some basic properties of exponentiation?
If we have a^b then 'a' is called base and 'b' is called power or exponent and we call it "a is raised to the power b" (this statement might change from text to text slightly).
Exponentiation(the process of raising some number to some power) have some basic rules as:
[tex]a^{-b} = \dfrac{1}{a^b}\\\\a^0 = 1 (a \neq 0)\\\\a^1 = a\\\\(a^b)^c = a^{b \times c}\\\\ a^b \times a^c = a^{b+c} \\\\^n\sqrt{a} = a^{1/n} \\\\(ab)^c = a^c \times b^c\\\\a^b = a^b \implies b= c \: \text{ (if a, b and c are real numbers and } a \neq 1 \: and \: a \neq -1 )[/tex]
The specified question is missing choices. They are:
A. [tex]32^{x/4}[/tex]
B. [tex](32/4)^x[/tex]
C. [tex]8 * 8^{x + 1}[/tex]
D. [tex]x^4[/tex]
E. [tex]\dfrac{32^x}{4^x}[/tex]
F. [tex]8 * 8^{x-1}[/tex]
The considered expression is: [tex]8^x[/tex]
Checking all the options one by one:
- A. [tex]32^{x/4}[/tex]
Assuming it is equivalent to [tex]8^x[/tex], then its value must be equal to value of [tex]8^x[/tex] for all x.
[tex](32)^{x/4}=8^x\\\\8^x = ( \: ^4\sqrt{32})^x\\\\8 = \: ^4\sqrt{32}\\8 \approx 2.37[/tex]
This is wrong. Thus, our assumption was wrong, and therefore, [tex]32^{x/4}[/tex] is not an equivalent expression to [tex]8^x[/tex]
- B. [tex](32/4)^x[/tex]
Assuming it is equivalent to [tex]8^x[/tex], then its value must be equal to value of [tex]8^x[/tex] for all x.
[tex](32/4)^x = 8^x\\\\8^x = 8^x[/tex]
This validates, so [tex](32/4)^x[/tex] is an equivalent expression to [tex]8^x[/tex]
- C. [tex]8 * 8^{x + 1}[/tex]
Assuming it is equivalent to [tex]8^x[/tex], then its value must be equal to value of [tex]8^x[/tex] for all x.
[tex]8 * 8^{x + 1} = 8^x\\\\8^{x+1+1} = 8^x\\8^{x+2} = 8^x\\x+2 = x\\2=0[/tex]
This is false statement. Thus, our assumption was wrong, and therefore, [tex]8 * 8^{x + 1}[/tex] is not an equivalent expression to [tex]8^x[/tex]
- D. [tex]x^4[/tex]
Assuming it is equivalent to [tex]8^x[/tex], then its value must be equal to value of [tex]8^x[/tex] for all x.
At x = 1, we get:
[tex]x^4 = 1^4 = 1[/tex]
But [tex]8^x = 8^1 = 8 \neq 1[/tex]
Thus, [tex]x^4[/tex] is not an equivalent expression to [tex]8^x[/tex]
- E. [tex]\dfrac{32^x}{4^x}[/tex]
[tex]\dfrac{32^x}{4^x} = (32/4)^x = 8^x[/tex]
Simplifying it, we get [tex]8^x[/tex], so it is an equivalent expression to [tex]8^x[/tex]
- F. [tex]8 * 8^{x-1}[/tex]
[tex]8 * 8^{x-1} = 8^{x-1+1} = 8^x[/tex]
Simplifying it, we get [tex]8^x[/tex], so it is an equivalent expression to [tex]8^x[/tex]
Thus, the expressions out of the specified expression which are equivalent to 8^x are given by:
- B. [tex](32/4)^x[/tex]
- E. [tex]\dfrac{32^x}{4^x}[/tex]
- F. [tex]8 * 8^{x-1}[/tex]
Learn more about exponentiation here:
https://brainly.com/question/26938318
