Solve each equation. See Examples 4–6. (√2)^(x+4) = 4^x

Question

Accepted Answer

Hello, everyone we are asked to solve for X. In the given equation, the equation we are given is the square root of three raised to the two X plus seventh power equals nine, raised to the negative X power. Our answer choices are a negative 7/ B X equals 76 C X equals 6/7 D X equals negative six sevens, rewriting our original equation again, I have the square root of three in parentheses. Raise to the two X plus seventh power equals nine. Raise the negative X power. Ideally here, I want to have both bases be the same number. So I can compare my exponents. I'm gonna start with a square out of three. I know that I can rewrite radical three as three. Raise to the one half power. So in parentheses, I'm rewriting the square to three as three, raised the one half power. And that's still gonna be raised to the two X plus seventh power. And that's still gonna be equal at this point to X or sorry. Nine raised to the negative X power. I recall that when I have a power raised to a power, I multiply my exponents and keep the base the same. So I'm gonna keep my three and I'm gonna multiply one half by two X plus seven. So one half multiplied by two X is X, one half multiplied by seven is gonna be plus seven halves. So at this step, I have three rays to the X plus seven halves. Power equals and I'm going to rewrite nine as three squared. So three squared is in parentheses and now it's raised to the negative X power. I'm gonna rewrite this again. So three raise to the X plus seven halves, power equals and then when I have three squared, raised to the negative X power, I again, when raising a power to a power multiply my exponents. So this will give me three raise to the negative two X power. And now that I have the base is the same, I can compare the exponents and set them equal to each other and solve for X. So I will have X plus seven halves equals negative two X. I'm gonna multiply the entire problem through by two to get rid of my fraction. And when I do that, I get two X plus seven equals negative four X solving for X, I'm gonna subtract two X from both sides. So I get seven equals negative six X. And to get the X by itself, I divide both sides by negative six and I end up with X equals negative 76. That is my answer and it matches answer choice a, have a nice day.

Solve each equation. See Examples 4–6. (√2)^(x+4) = 4^x

Key Concepts

Exponential Functions

Logarithms

Change of Base Formula

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