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Question: If we are an expression as \[{{4}^{x}}-{{4}^{x-1}}=24\], then \[{{\left( 2x \right)}^{x}}\] equals t...

If we are an expression as 4x−4x−1=24{{4}^{x}}-{{4}^{x-1}}=24, then (2x)x{{\left( 2x \right)}^{x}} equals to?

Explanation

Solution

At first find the value of x and then substitute it. At first, consider the equation and take 4x−1{{4}^{x-1}} common, then divide it by 3. After that write 4 as 22{{2}^{2}} and 8 as 23{{2}^{3}}. Then apply the principle that if bases are the same exponents should be equal to find the value of x.

Complete step-by-step solution:
In the question we are given an equation 4x−4x−1=24{{4}^{x}}-{{4}^{x-1}}=24 and we have to find the value of (2x)x{{\left( 2x \right)}^{x}}.
Before finding the value of (2x)x{{\left( 2x \right)}^{x}} we will first find the value of x.
Now as we are given that,
4x−4x−1=24{{4}^{x}}-{{4}^{x-1}}=24
So to proceed we have to take 4x−1{{4}^{x-1}} common so we can write as,
4x−1+1−4x−1=24{{4}^{x-1+1}}-{{4}^{x-1}}=24
Or, 4x−1(4−1)=24{{4}^{x-1}}\left( 4-1 \right)=24
So, the equation can be written as,
4x−1.3=24{{4}^{x-1}}.3=24
Now we will divide by 3 to both sides.
So, we get,
4x−1=243{{4}^{x-1}}=\dfrac{24}{3}
Or, 4x−1=8{{4}^{x-1}}=8
Now as we know that 4 = 22{{2}^{2}} and 8 = 23{{2}^{3}}. So, we can also write 4x−1{{4}^{x-1}} as 22(x−1){{2}^{2\left( x-1 \right)}} or 22x−2{{2}^{2x-2}}.
Hence, the equation can be written as,
22x−2=23{{2}^{2x-2}}={{2}^{3}}
Now as we know that if bases are the same exponents will be equal, so here bases are 2 which is the same on both sides, and here exponents are (2x - 2) and 3 which according to rule should be equal.
Hence, we can write,
2x–2=32x – 2 = 3
So, 2x=52x = 5
Hence, the value of x=52x=\dfrac{5}{2}.
Now as we know x so we will substitute in (2x)x{{\left( 2x \right)}^{x}} to get the answer which is (2×52)52{{\left( 2\times \dfrac{5}{2} \right)}^{\dfrac{5}{2}}} which is (5)52{{\left( 5 \right)}^{\dfrac{5}{2}}} or (5)2×(5)12=25×5{{\left( 5 \right)}^{2}}\times {{\left( 5 \right)}^{\dfrac{1}{2}}}=25\times \sqrt{5} or 25525\sqrt{5}.
Hence the value is 25525\sqrt{5}.

Note: After simplifying the equation to 22x−2=23{{2}^{2x-2}}={{2}^{3}}, we compared the bases and equalized the exponents so instead of this method we can take logarithms to both the sides and then proceed. Also, students after finding out value x miss that they have to find the value of what is asked so be careful about it.