https://file.aiquickdraw.com/mathbot/file/1727102880412muk9lqun.jpg

The image contains two equations that need to be solved for \( A \) and \( B \). The given equations are:

\[
x^A \cdot x^B = x^8
\]
and
\[
\frac{x^A}{x^B} = x^{12}
\]

### Step 1: Simplify the first equation
Using the law of exponents \( x^m \cdot x^n = x^{m+n} \), the first equation simplifies to:

\[
x^{A + B} = x^8
\]

By comparing exponents (since the bases are the same), we get:

\[
A + B = 8 \quad \text{(Equation 1)}
\]

### Step 2: Simplify the second equation
Using the law of exponents \( \frac{x^m}{x^n} = x^{m-n} \), the second equation simplifies to:

\[
x^{A - B} = x^{12}
\]

By comparing exponents again, we get:

\[
A - B = 12 \quad \text{(Equation 2)}
\]

### Step 3: Solve the system of equations
Now, we have the following system of equations:
1. \( A + B = 8 \)
2. \( A - B = 12 \)

To solve this, add both equations:

\[
(A + B) + (A - B) = 8 + 12
\]
\[
2A = 20
\]
\[
A = 10
\]

Now substitute \( A = 10 \) into the first equation \( A + B = 8 \):

\[
10 + B = 8
\]
\[
B = 8 - 10 = -2
\]

### Final Answer:
- \( A = 10 \)
- \( B = -2 \)

Would you like further explanation or have any other questions?

### Related Questions:
1. How do exponent rules apply to multiplication and division of terms?
2. What are other methods to solve systems of equations?
3. How can exponent equations be used in real-world applications?
4. What happens if you change the base of these exponent equations?
5. What are the limitations of solving exponent equations algebraically?

**Tip**: Always remember to apply exponent rules consistently when solving equations involving powers!

The following two equations are true. What are the values of A and B? x^A * x^B = x^8 and (x^A) / (x^B) = x^12

Learn how to solve for A and B in the equations x^A * x^B = x^8 and (x^A) / (x^B) = x^12 using exponent rules. This step-by-step solution covers the laws of exponents and system of equations, suitable for students in Grades 9-10.

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