Question

If$9^x - 9^{x -1} = 648$, then find the value of $x^x$

• A. 4
• B. 27
• C. 9
• D. 64

Answer 1

Answer:

B. 27

• Rule 1: Division of Exponents with Same Base

  • a^m / aⁿ = a^m-n

  • This rule is vital for transforming 9^x-1 into 9^x / 9¹ or 9^x / 9.

• Rule 2: Multiplication of Exponents with Same Base

  • a^m × aⁿ = a^m+n

  • This rule is used when combining terms like 9² × 9¹ to get 9³.

• Rule 3: Equating Powers

  • If a^x = a^y (where a is a positive real number not equal to 1), then x = y.

  • This rule allows you to solve for x once both sides of the equation are expressed with the same base.

Step-by-Step Solution Breakdown:

1. Simplify the Equation:
   The given equation is 9^x - 9^x-1 = 648. Applying the exponent rule a^m-n = a^m / aⁿ, we can rewrite 9^x-1 as 9^x / 9.

2. Rewrite the Equation:
   The equation becomes 9^x - (9^x / 9) = 648.

3. Factor out Common Term:
   Factor out 9^x from both terms on the left side: 9^x (1 - 1/9) = 648.

4. Simplify the Bracketed Term:
   Calculate the value inside the parenthesis: 1 - 1/9 = (9 - 1)/9 = 8/9.

5. Equation after Simplification:
   So, the equation reduces to 9^x (8/9) = 648.

6. Isolate 9^x:
   Multiply both sides by 9/8 to isolate 9^x: 9^x = 648 × (9/8).

7. Perform the Division and Multiplication:
   First, divide 648 by 8: 648 ÷ 8 = 81.
   Then, multiply the result by 9: 81 × 9 = 729.
   So, 9^x = 729.

8. Express both Sides with the Same Base:
   Recognize that 729 is a power of 9. We know 9¹ = 9, 9² = 81, and 9³ = 81 × 9 = 729.
   Thus, 9^x = 9³.

9. Solve for x:
   Using the rule that if bases are equal, exponents must be equal, we get x = 3.

10. Calculate x^x:
    The problem asks for the value of x^x. Substitute x = 3 into this expression: 3³.

11. Final Result:
    3³ = 3 × 3 × 3 = 27.