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Algebra Level 2

If $x$ satisfy the equation $4^x-4^{x-1}=24$ , find the value of ${(2x)}^x$ .

5\sqrt{5}

25\sqrt{5}

125

None of these choices

6 solutions

Ajey Devadiga
May 11, 2016

4^x - 4^(x-1) = 24
=(2^x)^2 - 4^x/4 = 24
=(2^x)^2 - (2^x)^2/4 =24

Let 2^x=a then
=a^2 - a^2/4 =24
=3a^2/4 =24
=3a^2 = 96
=a^2 = 32

But a=2^x then
=(2^x)^2 = 32
=2^2x = 2^5 then
=2x = 5
=x= 5/2

And value of
=(2x)^x
=(2×5/2)^5/2
=5^(5/2) this is equal to 25×sqrt(5)

I will help you to modify your solution by using the $\LaTeX$ codes

$4^x - 4^{x-1} = 24$

$\Rightarrow (2^x)^2 - 4^{x/4} = 24$

$\Rightarrow (2^x)^2 - (2^x)^{2/4} =24$

Let $2^x=a$ then $a^2 - a^2/4 =24$

$3a^2/4 =24$

$3a^2 = 96$

$=a^2 = 32$

But $a=2^x$ then $2^{x^{2}} = 32 =2^{2x} = 2^5$ then $=2x = 5 =x= 5/2$

And value of $= 2x^x =(2×5/2)^{5/2} =5^{5/2}$ this is equal to $25\sqrt{5}$ ?

This is based on your solution by using your natural solution.

A Former Brilliant Member - 5 years, 1 month ago

that is better and good solution...+1

Ayush G Rai - 5 years, 1 month ago

$\ddot \smile$

A Former Brilliant Member - 5 years, 1 month ago

Thank you cedie camomot

Ajey Devadiga - 5 years, 1 month ago

$\text{Your welcome} \ddot \smile \space \text{practice} \space L_A^TeX \space \text{soon}$

A Former Brilliant Member - 5 years, 1 month ago

Brilliant..!! the answer is now much More attractive and easy to understand.+1

Rishabh Tiwari - 5 years, 1 month ago

good solution but use latex to be more clear...+1

Ayush G Rai - 5 years, 1 month ago

Thanks for sharing your approach. (+1) You can insert a line break by leaving three spaces at the end of the line. I have edited your solution accordingly.

Pranshu Gaba - 5 years, 1 month ago

thank you very much

Ayush G Rai - 5 years, 1 month ago

Nice sol. Did it the same way..!

Rishabh Tiwari - 5 years, 1 month ago

Great solution , same way, +1 :)

Novril Razenda - 4 years, 11 months ago

Hung Woei Neoh
May 12, 2016

$4^x - 4^{x-1} = 24\\ 4^x - \dfrac{4^x}{4} = 24\\ 4^x \left( 1 - \dfrac{1}{4} \right) = 24\\ 4^x \left(\dfrac{3}{4}\right) = 24\\ 4^x = 24 \times \dfrac{4}{3}\\ 4^x = 32\\ 2^{2x} = 2^5\\ 2x=5\\ x=\dfrac{5}{2}\\ (2x)^x\\ =\left(2\times \dfrac{5}{2} \right)^{\frac{5}{2}}\\ =5^{2+\frac{1}{2}}\\ =5^2 \times 5^{\frac{1}{2}}\\ =\boxed{25\sqrt{5}}$

love this answer, it shows it so clearly

Nicole Tzan - 5 years ago

Thanks $\ddot\smile$

Hung Woei Neoh - 5 years ago

A Former Brilliant Member
May 11, 2016

$2^{2x}-2^{2x-2}=2^3×3$

$\dfrac{2^{2x}-2^{2x-2}}{2^3}=3$

$2^{2x}\left(\dfrac{1}{8}-\dfrac{1}{32}\right)=3$

$2^{2x}=32$

$x=\dfrac{5}{2}$

$\therefore (2x)^x=\left(2×\dfrac{5}{2}\right)^{\frac{5}{2}}=\boxed{25\sqrt{5}}$

in step 2:how did $2^{2x-2}$ become $2^{2x-5}$ ?

Ayush G Rai - 5 years, 1 month ago

It was typo...thanks!!

A Former Brilliant Member - 5 years, 1 month ago

it is was a pleasure to help you as you had a different answer.

Ayush G Rai - 5 years, 1 month ago

Damien G
May 12, 2016

Maybe a faster solution here: $4^x-4^{x-1}=28$

$\Rightarrow 3*4^{x-1}=3*8$

$\Rightarrow 2^ {2*(x-1)}=2^3$

$\Rightarrow 2x-2=3$

$\Rightarrow x=5/2$

$\Rightarrow (2x)^x=5^{5/2}=25*\sqrt{5}$

Your solution is a mess.

The original equation is $4^x - 4^{x-1} = 24$

And how did you get $3 \times 4^{x-1} = 3 \times 8$ ?

Hung Woei Neoh - 5 years, 1 month ago

I agree with you.that was a wrong solution from damien G

Ayush G Rai - 5 years, 1 month ago

He means 24 = 3 x 8 and $4^x - 4^{x-1} = (4-1)(4^{x-1}) = 3 * (4^{x-1})$

Ken Gene Quah - 5 years, 1 month ago

Ah, I see. Now I understand

Hung Woei Neoh - 5 years, 1 month ago

now i understood what he meant to say. $4^x-4^{x-1}=4^x\times4^{-1}=$ ( $4^x)$ ( $1-\frac{1}{4})$ =( $4^x)$ ( $\frac{3}{4})$
$=3\times$ ( $4^{x-1})$

Ayush G Rai - 5 years, 1 month ago

now i understood what he meant to say. $4^x-4^{x-1}=4^x\times4^{-1}=$ ( $4^x)$ ( $1-\frac{1}{4})$ =( $4^x)$ ( $\frac{3}{4})$
$=3\times$ $4^{x-1}$

Ayush G Rai - 5 years, 1 month ago

Prince Loomba
May 28, 2016

Take out $4^{x-1}$ common and cancel 3 from both sides to get x=2.5 and then put in the required answer

good observation

Ayush G Rai - 5 years ago

Why not +1

Prince Loomba - 5 years ago

ok,I have given!!!

Ayush G Rai - 5 years ago

@Ayush G Rai – Thanx ayush

Prince Loomba - 5 years ago

Patience Patience
May 12, 2016

=4^x(1-.25)=4^2.5>>>>>>> x=2.5 >>>>

can you be clear in writing your solution by using latex.

Ayush G Rai - 5 years, 1 month ago

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