MATH SOLVE

5 months ago

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

Accepted Solution

A:

ANSWER

x = 5/4

EXPLANATION

[tex]2^x \cdot 2^{x-2} = \sqrt{2}[/tex]

Note that [tex]\sqrt{a} = a^{\frac{1}{2}}[/tex] so [tex]\sqrt{2} = 2^{\frac{1}{2} }[/tex]

Note that on the left-hand side, we can use exponent properties for multiplying two powers of the same base together: [tex]a^x \cdot a^y = a^{x+y} [/tex]

[tex]\begin{aligned} 2^x \cdot 2^{x-2} &= \sqrt{2} \\ 2^{x + (x-2)} &= 2^{\frac{1}{2}} \\ 2^{2x - 2} &= 2^{\frac{1}{2}} \end{aligned}[/tex]

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

[tex]\begin{aligned} 2^{2x - 2} &= 2^{\frac{1}{2}} \\ 2x - 2 &= \tfrac{1}{2} \\ 2x &= \tfrac{1}{2} + 2 \\ 2x &= \tfrac{5}{2} \\ x &= \tfrac{5}{4} \end{aligned}[/tex]

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

x = 5/4

EXPLANATION

[tex]2^x \cdot 2^{x-2} = \sqrt{2}[/tex]

Note that [tex]\sqrt{a} = a^{\frac{1}{2}}[/tex] so [tex]\sqrt{2} = 2^{\frac{1}{2} }[/tex]

Note that on the left-hand side, we can use exponent properties for multiplying two powers of the same base together: [tex]a^x \cdot a^y = a^{x+y} [/tex]

[tex]\begin{aligned} 2^x \cdot 2^{x-2} &= \sqrt{2} \\ 2^{x + (x-2)} &= 2^{\frac{1}{2}} \\ 2^{2x - 2} &= 2^{\frac{1}{2}} \end{aligned}[/tex]

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

[tex]\begin{aligned} 2^{2x - 2} &= 2^{\frac{1}{2}} \\ 2x - 2 &= \tfrac{1}{2} \\ 2x &= \tfrac{1}{2} + 2 \\ 2x &= \tfrac{5}{2} \\ x &= \tfrac{5}{4} \end{aligned}[/tex]

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