Math  /  Algebra

Question3-37. Using the laws of exponents to solve for A:64x+2=A4xA: 6 \cdot 4^{x+2}=A \cdot 4^{x}.

Studdy Solution

STEP 1

What is this asking? We need to find the value of AA that makes the equation 64x+2=A4x6 \cdot 4^{x+2} = A \cdot 4^x true for all values of xx. Watch out! Don't forget your exponent rules!
Also, don't be intimidated by the xx!

STEP 2

1. Expand and Simplify
2. Isolate AA

STEP 3

Remember that 4x+24^{x+2} is the same as 4x424^x \cdot 4^2.
Why? Because when you multiply exponential expressions with the same **base**, you *add* the **exponents**!
So, we can rewrite our equation as 64x42=A4x6 \cdot 4^x \cdot 4^2 = A \cdot 4^x.

STEP 4

Now, let's calculate 424^2.
That's just 44=164 \cdot 4 = \textbf{16}.
So, our equation becomes 64x16=A4x6 \cdot 4^x \cdot \textbf{16} = A \cdot 4^x.

STEP 5

Let's multiply the **constants** 66 and 16\textbf{16} together. 616=966 \cdot \textbf{16} = \textbf{96}.
Now our equation looks much simpler: 964x=A4x\textbf{96} \cdot 4^x = A \cdot 4^x.

STEP 6

To get AA by itself, we can divide both sides of the equation by 4x4^x.
This gives us 964x4x=A4x4x\frac{\textbf{96} \cdot 4^x}{4^x} = \frac{A \cdot 4^x}{4^x}.

STEP 7

Since 4x4x=1\frac{4^x}{4^x} = 1, our equation simplifies to 961=A1\textbf{96} \cdot 1 = A \cdot 1, which means 96=A\textbf{96} = A.
We found it!

STEP 8

A=96A = \textbf{96}.

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