Solve for x

2^x+1 - 2^x = 112

Put this in exponent form:
(3square root of x) ( square root of 3 cubed)

Put into equation form:
Horizontal stretch of 2, up 3, left 6, reflection on x-axis.

Please help me with these three so I can work on the rest using these as examples.

Thanks so much for your help!!!!:)

To solve the equation 2^(x+1) - 2^x = 112, we can use the properties of exponents to simplify and solve for x.

Step 1: Simplify the left side of the equation by combining the two exponents:

2^(x+1) - 2^x = 2^x * 2^1 - 2^x = 2*2^x - 2^x = 2^x(2-1) = 2^x.

Now, the equation becomes: 2^x = 112.

Step 2: Convert both sides of the equation to a common base. Let's choose base 2 since we have already been working with powers of 2 here.

2^x = 112 can be written as 2^x = 2^7, since 112 is equal to 2 raised to the power of 7.

Step 3: Set the exponents equal to each other and solve for x:

x = 7.

Therefore, x = 7 is the solution to the equation 2^(x+1) - 2^x = 112.

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To put (3√x)(√3)^3 into exponent form, we can simplify the expression:

First, we apply the cube root to x inside the parentheses.

(3√x)(√3)^3 = (x^(1/3))(3)^(3/2) = x^(1/3) * (3^(3/2)).

Next, we simplify the cube root and the square root separately.

x^(1/3) = ∛x,

3^(3/2) = √(3^3) = √27.

Putting it all together, the expression becomes:

(3√x)(√3)^3 = ∛x * √27.

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To put the given description "horizontal stretch of 2, up 3, left 6, reflection on x-axis" into equation form, let's break it down into individual transformations:

1. Horizontal stretch of 2:
This can be written as f(2x) for a given function f(x).

2. Up 3:
This can be written as f(x) + 3.

3. Left 6:
This can be written as f(x + 6).

4. Reflection on the x-axis:
This can be written as -f(x).

Now, combining all the transformations, the equation form becomes:

-f(x + 6) + 3.

This equation represents a function that has been horizontally stretched by a factor of 2, moved up 3 units, shifted left 6 units, and reflected on the x-axis.