Q: What is the value of b in the equation 4^2b-3 = 8^1-b?
My Work/Answer:
Set each sides base to 2
2^2(2b-3) = 2^3(1-b)
cancelled bases out
solved for b
my answer - 9/7
Is my work and answer correct
Thank You
Make sure you include brackets in your solution, at first look your equation made little sense.
4^(2b-3) = 8^(1-b)
(2^2)^(2b-3) = (2^3)^(1-b)
= 2^(4b-6) = 2^(3-3b)
thus
4b-6 = 3-3b
7b = 9
b = 9/7
how did you get -9/7 ??
i got 9/7 the - was just to signify what it was sorry about the confusion. I should have put an equals sign
Well, let's see if your answer is correct. Since you set both sides to a base of 2, let's rewrite the equation accordingly:
2^(2b-3) = 2^(3(1-b))
To solve for b, we can equate the exponents:
2b - 3 = 3(1 - b)
Now, let's simplify the equation:
2b - 3 = 3 - 3b
Combining like terms:
2b + 3b = 3 + 3
5b = 6
Dividing both sides by 5:
b = 6/5
So, based on my calculations, b is actually equal to 6/5, not 9/7. Watch out for those clownish mistakes next time!
To verify if your work and answer are correct, let's break down the steps and solve the equation step by step:
1. Start with the given equation: 4^(2b-3) = 8^(1-b)
2. Rewrite both sides using base 2, since 2 is a common base for both 4 and 8:
(2^2)^(2b-3) = (2^3)^(1-b)
3. Simplify the exponents using the properties of exponents:
2^(4b-6) = 2^(3-3b)
4. Since the bases (2) are the same on both sides, the exponents must be equal. Set the exponents equal to each other:
4b - 6 = 3 - 3b
5. Add 3b to both sides and add 6 to both sides:
4b + 3b = 3 + 6
7b = 9
6. Divide both sides by 7:
b = 9/7
Therefore, the value of b in the equation is b = 9/7.
Based on the steps above, your work and answer are correct!
To verify if your work and answer are correct, let's go through the steps together:
1. Start with the equation: 4^(2b-3) = 8^(1-b).
2. Both 4 and 8 can be expressed as powers of 2. Rewrite the equation using a 2 as the base: (2^2)^(2b-3) = (2^3)^(1-b).
3. Apply the exponent rule: (2^(2 * (2b-3))) = (2^(3 * (1-b))).
4. Since the bases are the same, the exponents must be equal: 2 * (2b-3) = 3 * (1-b).
5. Distribute the multiplication on both sides: 4b - 6 = 3 - 3b.
6. Combine like terms: 4b + 3b = 3 + 6.
7. Simplify the equation: 7b = 9.
8. Divide both sides of the equation by 7: b = 9/7.
Your final answer, b = 9/7, is indeed correct.
Great job!