Solving for 8^x * 2^y Given the Equation 3x - y 12

When dealing with algebraic expressions involving exponents, it's important to use the properties of exponents to simplify the expressions. In this problem, we will solve for the expression 8x * 2y given that 3x - y 12.

Step 1: Express 8x in Terms of Base 2

Since 8 can be expressed as 23, we can write:

8x (23)x 23x

Step 2: Rewrite the Expression 8x * 2y

Now, we can rewrite the original expression:

8x * 2y 23x * 2y

Using the property of exponents a^m * a^n a^{m n}, we get:

23x * 2y 2^{3x y}

Step 3: Express y in Terms of x

Given the equation 3x - y 12, we can express y as:

y 3x - 12

Step 4: Substitute y into the Expression

Now, we substitute the value of y into the expression:

2^{3x y} 2^{3x 3x - 12}

Simplifying the exponent:

3x 3x - 12 6x - 12

This means:

2^{6x - 12}

We can also factor out a 6 from the exponent:

2^{6x - 12} 2^{6x - 2}

Conclusion

The expression 8^x * 2^y simplifies to:

8^x * 2^y 2^{6x - 2}

Without a specific value for x, we cannot compute a numerical value for 8^x * 2^y. However, the expression 2^{6x - 2} represents the relationship based on the original equation.

Additional Examples

Consider the following additional examples:

x y 8^x 2^y 8^x * 2^y 4 0 256 1 256 2 -6 4 1/64 1/16

These examples demonstrate the variability in the values of x and y, and the resulting expression 8^x * 2^y.