If 44x = 1600, what is the value of (4x–1)2?

For this part of the solution: (4x–1)^2 = 4^2x– 2
If we further simplify we get the expression 4^2x/4^2

Is there a rule about getting from 4^2x-2 to 4^2x/4^2?

Thank you!


It is tempting to express both sides of the equation 44x = 1600 as powers of 4 and to try and solve for x. However, if we do that, we get a power of five on the right side as well:
44x = 16 × 100
44x = 42 × 4 × 25
44x = 43 × 52
It becomes clear that x is not an integer and that we can’t solve the question this way.

Let’s try manipulating the expression about which we are being asked.
(4x–1)2 = 42x– 2
If we further simplify we get the expression 42x/42
To solve this expression, all we need is to find the value of 42x

Now let’s look back at our original equation. If 44x = 1600, we can find the value of 42x by taking the square root of both sides of the equation. Taking the square root of an exponential expression is tantamount to halving its exponent.

Since the question asks for 42x/42, the answer is 40/42, which simplifies to 40/16 or 5/2.

The correct answer is D. (5/2)

I am missing something here.

My apologies, the powers did not show up. Please see below. Thank you!

[quote="Anonymous"]If 4^4x = 1600, what is the value of (4^x–1)^2?

For this part of the solution: (4^x–1)^2 = 4^2x– 2
If we further simplify we get the expression 4^2x/4^2

Is there a rule about getting from 4^2x-2 to 4^2x/4^2?

Thank you!

Solution:
It is tempting to express both sides of the equation 4^4x = 1600 as powers of 4 and to try and solve for x. However, if we do that, we get a power of five on the right side as well:
4^4x = 16 × 100
4^4x = 42 × 4 × 25
4^4x = 43 × 52
It becomes clear that x is not an integer and that we can’t solve the question this way.

Let’s try manipulating the expression about which we are being asked.
(4^x–1)^2 = 42^x– 2
If we further simplify we get the expression 4^2x/4^2
To solve this expression, all we need is to find the value of 4^2x

Now let’s look back at our original equation. If 4^4x = 1600, we can find the value of 42x by taking the square root of both sides of the equation. Taking the square root of an exponential expression is tantamount to halving its exponent.

Since the question asks for 4^2x/4^2, the answer is 40/4^2, which simplifies to 40/16 or 5/2.

The correct answer is D. (5/2)

Why does taking the square root of 4^4x = 4^2x and not 2^2x? Thanks

Try with some real numbers so you can see what's going on.

What's the square root of 4^4?
4^4 = 4*4*4*4
SQRT(4*4*4*4) = 4*4 = 16
16 is the equivalent of 4^2 or 2^4 - both are right. I can write the answer either way, just depending what I want the base to be.

If you start with 4^4x, write out the steps using your exponent rules:
4^4x =
(4^x)(4^x)(4^x)(4^x)
then, if I take the square root above, I cross off one 4^x for every two 4^x terms present, leaving me with:
(4^x)(4^x) =
4^2x