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Difficulty: MediumExponential Functions and Equations

If 4a⋅8b=3254^{a} \cdot 8^{b} = 32^{5} and a+b=9a + b = 9, what is the value of bb?

  1. A
    2
  2. B
    5
  3. 7Answer
  4. D
    11

Answer

7
By converting all bases to 2, the equation 4a⋅8b=3254^a \cdot 8^b = 32^5 becomes (22)a⋅(23)b=(25)5(2^2)^a \cdot (2^3)^b = (2^5)^5, which simplifies to 22a+3b=2252^{2a+3b} = 2^{25}. Equating the exponents gives the linear equation 2a+3b=252a + 3b = 25. Since we are given that a+b=9a + b = 9, we can multiply this equation by 2 to get 2a+2b=182a + 2b = 18. Subtracting the two equations yields b=7b = 7.

Step-by-Step Solution

1
Express each base in the equation 4a⋅8b=3254^a \cdot 8^b = 32^5 as a power of 2.
Since 4=224 = 2^2, 8=238 = 2^3, and 32=2532 = 2^5, the equation becomes (22)a⋅(23)b=(25)5(2^2)^a \cdot (2^3)^b = (2^5)^5.
To solve exponential equations with different bases, it is helpful to express all terms using a common base.
2
Apply the power-of-a-power property (xm)n=xmn(x^m)^n = x^{mn} and the product-of-powers property xm⋅xn=xm+nx^m \cdot x^n = x^{m+n} to simplify both sides.
The left side simplifies to 22a⋅23b=22a+3b2^{2a} \cdot 2^{3b} = 2^{2a+3b}, and the right side simplifies to 2252^{25}. The equation is now 22a+3b=2252^{2a+3b} = 2^{25}.
Simplifying the expressions allows us to equate the exponents.
3
Set the exponents equal to each other to form a linear equation.
2a+3b=252a + 3b = 25.
If two exponential expressions with the same base are equal, their exponents must be equal.
4
Solve the system of equations consisting of 2a+3b=252a + 3b = 25 and a+b=9a + b = 9.
Multiplying the second equation by 2 gives 2a+2b=182a + 2b = 18. Subtracting this from 2a+3b=252a + 3b = 25 yields (2a+3b)−(2a+2b)=25−18(2a + 3b) - (2a + 2b) = 25 - 18, which simplifies to b=7b = 7.
Eliminating one variable allows us to solve for the other variable directly.

Key Concept

Solving exponential equations by converting terms to a common base and solving the resulting system of linear equations.

Alternative Method

Substitute the answer options for bb back into the equations. If b=7b = 7, then a=9−7=2a = 9 - 7 = 2. Plugging these values into the left side of the exponential equation gives 42⋅87=16⋅(23)7=24⋅221=2254^2 \cdot 8^7 = 16 \cdot (2^3)^7 = 2^4 \cdot 2^{21} = 2^{25}. The right side is 325=(25)5=22532^5 = (2^5)^5 = 2^{25}. Since both sides are equal, 7 is the correct answer.
Estimated Time:1m 30s
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Topics

Exponential Functions and Equations

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