Question

Solve exponential equation: 2^x * 2^x-2= √2 (Picture included)

Answer 1

Hi there!

Th' Eqⁿ is :-


`2^x × 2^(x-2) = √(2)`


`\begin{aligned} 2^x \cdot 2^(x-2) &= √(2) \\ 2^(x + (x-2)) &= 2^{(1)/(2)} \\ 2^(2x - 2) &= 2^{(1)/(2)} \end{aligned}`

Equating the Exponent :-

Cuz' both sides of the equation have th' same base with no other terms.


`\begin{aligned} 2^(2x - 2) &= 2^{(1)/(2)} \\ 2x - 2 &= \tfrac{1}{2} \\ 2x &= \tfrac{1}{2} + 2 \\ 2x &= \tfrac{5}{2} \\ x &= \tfrac{5}{4} \end{aligned}`

Hence,
The required answer is x =
`\frac {5}{4}`

~ Hope it helps!

Answer 2

ANSWER

x = 5/4

Step-by-step explanation


`2^x \cdot 2^(x-2) = √(2)`

Note that
`√(a) = a^{(1)/(2)}` so
`√(2) = 2^{(1)/(2) }`
Note that on the left-hand side, we can use exponent properties for multiplying two powers of the same base together:
`a^x \cdot a^y = a^(x+y)`


`\begin{aligned} 2^x \cdot 2^(x-2) &= √(2) \\ 2^(x + (x-2)) &= 2^{(1)/(2)} \\ 2^(2x - 2) &= 2^{(1)/(2)} \end{aligned}`

We can now equate the exponents because both sides of the equation are of the same base with no other terms.


`\begin{aligned} 2^(2x - 2) &= 2^{(1)/(2)} \\ 2x - 2 &= \tfrac{1}{2} \\ 2x &= \tfrac{1}{2} + 2 \\ 2x &= \tfrac{5}{2} \\ x &= \tfrac{5}{4} \end{aligned}`

The answer is x = 5/4. We can confirm this by using this value in the original equation to get a true statement.