If $4^{x+6}=8^{x+2}=2^{y}$ , then find the values of $x$ andy.

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Algebra 2

Composition of linear functions: find a value

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Q. If

4^{x+6}=8^{x+2}=2^{y}

, then find the values of

x

andy.

Expressing $8$ as Power of $4$ : First, let's express $8$ as a power of $4$ , since $8$ is $2^3$ and $4$ is $2^2$ , we can write $8$ as $(2^2)^{3/2}$ which is $4$ $0$ .
Equating Powers of $4$ : Now, let's equate the powers of $4$ , so we have $4^{(x+6)} = 4^{\left(\frac{3}{2}\right)(x+2)}$ .
Setting Exponents Equal: Since the bases are the same, we can set the exponents equal to each other: $x+6 = \left(\frac{3}{2}\right)(x+2)$ .
Solving for x: Let's solve for x: $x+6 = \left(\frac{3}{2}\right)x + 3$ .
Finding $y$ : Subtract $\frac{3}{2}x$ from both sides: $x - \frac{3}{2}x + 6 = 3$ .
Finding $y$ : Subtract $\frac{3}{2}x$ from both sides: $x - \frac{3}{2}x + 6 = 3$ .Combine like terms: $\frac{1}{2}x + 6 = 3$ .
Finding y: Subtract $(\frac{3}{2})x$ from both sides: $x - (\frac{3}{2})x + 6 = 3$ .Combine like terms: $(\frac{1}{2})x + 6 = 3$ .Subtract $6$ from both sides: $(\frac{1}{2})x = -3$ .
Finding $y$ : Subtract $\frac{3}{2}x$ from both sides: $x - \frac{3}{2}x + 6 = 3$ .Combine like terms: $\frac{1}{2}x + 6 = 3$ .Subtract $6$ from both sides: $\frac{1}{2}x = -3$ .Multiply both sides by $2$ to solve for $x$ : $x = -6$ .
Finding y: Subtract $(\frac{3}{2})x$ from both sides: $x - (\frac{3}{2})x + 6 = 3$ .Combine like terms: $(\frac{1}{2})x + 6 = 3$ .Subtract $6$ from both sides: $(\frac{1}{2})x = -3$ .Multiply both sides by $2$ to solve for $x$ : $x = -6$ .Now we have the value of $x$ , let's find $y$ by using the equation $x - (\frac{3}{2})x + 6 = 3$ $0$ .
Finding y: Subtract $(\frac{3}{2})x$ from both sides: $x - (\frac{3}{2})x + 6 = 3$ .Combine like terms: $(\frac{1}{2})x + 6 = 3$ .Subtract $6$ from both sides: $(\frac{1}{2})x = -3$ .Multiply both sides by $2$ to solve for x: $x = -6$ .Now we have the value of x, let's find y by using the equation $2^{y} = 4^{(x+6)}$ .Substitute $x = -6$ into the equation: $2^{y} = 4^{(-6+6)}$ .
Finding y: Subtract $(\frac{3}{2})x$ from both sides: $x - (\frac{3}{2})x + 6 = 3$ .Combine like terms: $(\frac{1}{2})x + 6 = 3$ .Subtract $6$ from both sides: $(\frac{1}{2})x = -3$ .Multiply both sides by $2$ to solve for x: $x = -6$ .Now we have the value of x, let's find y by using the equation $2^{y} = 4^{x+6}$ .Substitute $x = -6$ into the equation: $2^{y} = 4^{-6+6}$ .Simplify the exponent: $2^{y} = 4^{0}$ .
Finding y: Subtract $(3/2)x$ from both sides: $x - (3/2)x + 6 = 3$ .Combine like terms: $(1/2)x + 6 = 3$ .Subtract $6$ from both sides: $(1/2)x = -3$ .Multiply both sides by $2$ to solve for $x$ : $x = -6$ .Now we have the value of $x$ , let's find $y$ by using the equation $x - (3/2)x + 6 = 3$ $0$ .Substitute $x = -6$ into the equation: $x - (3/2)x + 6 = 3$ $2$ .Simplify the exponent: $x - (3/2)x + 6 = 3$ $3$ .Since any number to the power of $x - (3/2)x + 6 = 3$ $4$ is $x - (3/2)x + 6 = 3$ $5$ , we have $x - (3/2)x + 6 = 3$ $6$ .
Finding y: Subtract $(\frac{3}{2})x$ from both sides: $x - (\frac{3}{2})x + 6 = 3$ .Combine like terms: $(\frac{1}{2})x + 6 = 3$ .Subtract $6$ from both sides: $(\frac{1}{2})x = -3$ .Multiply both sides by $2$ to solve for x: $x = -6$ .Now we have the value of x, let's find y by using the equation $2^{y} = 4^{x+6}$ .Substitute $x = -6$ into the equation: $2^{y} = 4^{-6+6}$ .Simplify the exponent: $2^{y} = 4^{0}$ .Since any number to the power of $0$ is $1$ , we have $2^{y} = 1$ .Now, let's express $1$ as a power of $2$ , which is $x - (\frac{3}{2})x + 6 = 3$ $0$ , so $x - (\frac{3}{2})x + 6 = 3$ $1$ .