Question

If $4^{(x -y) }= 64$ and $4^{(x + y) }= 1024$, then find the value of x.

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer:

D. 4

Convert Numbers to a Common Base

• The given equations are $4^{(x - y)} = 64$ and $4^{(x + y)} = 1024$.

• The first crucial step is to express the numbers on the right-hand side (64 and 1024) as powers of the base 4.

• We know that $64 = 4 \times 4 \times 4 = 4^3$.

• Similarly, $1024 = 4 \times 4 \times 4 \times 4 \times 4 = 4^5$.

Equate the Exponents

• Now, substitute these values back into the original equations:

  • Equation 1 becomes: $4^{(x - y)} = 4^3$

  • Equation 2 becomes: $4^{(x + y)} = 4^5$

• According to the law of exponents, if $a^m = a^n$ (where $a \ne 0, 1, -1$), then $m = n$.

• Applying this rule, we can equate the exponents:

  • New Equation 3: $x - y = 3$

  • New Equation 4: $x + y = 5$

Solve the System of Linear Equations

• We now have a simple system of two linear equations with two variables (x and y).

• To find 'x', the easiest method is elimination. Add Equation 3 and Equation 4:

  • $(x - y) + (x + y) = 3 + 5$

  • $x - y + x + y = 8$

  • $2x = 8$

• Finally, solve for 'x': $x = \frac{8}{2} = 4$.