Which of the following expressions are equivalent to 256^x? Select all that apply.

a. 8^2x
b. 4^4x
c. (64^x)^2
d. (16^x)^2
e. 2^8x​

Answers

Answer 1

Answer:

d

Step-by-step explanation:

i did it on eduhguenity

Answer 2

The solutions which satisfy the equation A = 256ˣ are

Option B. 4⁴ˣ

Option D. 16²ˣ

Option E. 2⁸ˣ

What is an Equation?

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is given by A = 256ˣ

a)

The value of equation B = 8^2x

B = 8²ˣ

On simplifying the equation , we get

B = 8²⁽ˣ⁾

B = 64ˣ

And the value of B ≠ A

b)

The value of equation B = 4^4x

B = 4⁴ˣ

On simplifying the equation , we get

B = 4⁴⁽ˣ⁾

B = 256ˣ

And the value of B = A

c)

The value of equation B = (64^x)^2

B = (64ˣ)²

On simplifying the equation , we get

B = 64²⁽ˣ⁾

B = 4096ˣ

And the value of B ≠ A

d)

The value of equation B = (16^x)^2

B = (16ˣ)²

On simplifying the equation , we get

B = 16²⁽ˣ⁾

B = 256ˣ

And the value of B = A

e)

The value of equation B = 2^8x​

B = (2⁸)ˣ

On simplifying the equation , we get

B = 256⁽ˣ⁾

And the value of B = A

Hence , the equations which satisfy are

Option B. 4⁴ˣ

Option D. 16²ˣ

Option E. 2⁸ˣ

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Answers

Answer:  39

==============================================

Explanation:

Let n be the number we want to find. We want n to be as small as possible, but also be a positive integer. Intuitively, we can see that n cannot be smaller than 16; otherwise, we don't meet the remainder requirements.

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-----------

We have these two equations

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Apply substitution and do a bit of rearranging like so

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The goal from here is to find the smallest positive integers x and y that make that equation true. We have a few options here and they are

Guess and check: We have a small sample size to work with so it shouldn't take too long. Make a table of xy values where you have x along the top row and y along the left column. Then plug each x,y pair into the equation above to see if you get a true statement or not. Again, keep in mind that x and y are positive integers. Graphing: Graph the line 3x-4y = 1, which is the same as y = (3/4)x - 1/4 and note where the line lands on a lattice point. Focus on the upper right quadrant of the graph. This quadrant is above the x axis and to the right of the y axis.Extended Euclidean Algorithm: This method is the most efficient, but it's only useful if your teacher has gone over it.

Whichever method you use, you should find that (x,y) = (3,2) is the point we want.

Note how:

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We know we hit the smallest value of n because (x,y) was made to be the smallest positive integer solution to 3x-4y = 1. There are infinitely many positive integer (x,y) solutions to 3x-4y = 1, which in turn means there are infinitely many numbers n that satisfy the remainder conditions (but n is not the smallest possible in those cases).


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