how to simplify:
((9^x * 3^2 * (1/3^-x)) - 27^x ) / 3^(3x) * 9
To simplify the given expression:
Step 1: Simplify individual terms
- Simplify 3^2 to 9: ((9^x * 9 * (1/3^-x)) - 27^x) / 3^(3x) * 9
Step 2: Simplify exponents
- Use the properties of exponents to simplify the expression further.
- For any base a, a^m * a^n = a^(m+n)
- For any base a, a^m / a^n = a^(m-n)
- Simplify (9^x * 9) / 3^-x to 9^(x+1) * 3^x: (9^(x+1) * (1/3^-x) - 27^x) / 3^(3x) * 9
Step 3: Simplify 27^x to (3^3)^x
- 27 equals 3^3, so replace 27^x by (3^3)^x: (9^(x+1) * (1/3^-x) - (3^3)^x) / 3^(3x) * 9
Step 4: Simplify terms with the same base
- Combine like terms by applying the properties of exponents:
- For any base a, a^m / a^n = a^(m-n)
- For any base a, a^m * a^n = a^(m+n)
- Simplify 9^(x+1) / 3^-x to 9^(x+1+x) = 9^(2x+1): (9^(2x+1) - (3^3)^x) / 3^(3x) * 9
Step 5: Simplify the denominator
- Use the property of exponents to bring 9 to the denominator:
- For any base a, a^m / a^n = a^(m-n)
- Simplify 9 * 3^-3x to 3^(2-3x): (9^(2x+1) - 3^(3x)*(3^3)^x) / 3^(3x) * 3^(2-3x)
Step 6: Combine terms with the same base
- Combine like terms by applying the properties of exponents:
- For any base a, a^m * a^n = a^(m+n)
- Simplify 3^(2-3x) * (3^3)^x to 3^(2-3x + 3x) = 3^2: (9^(2x+1) - 3^(3x+3x)) / 3^(3x) * 3^2
Step 7: Simplify further
- Simplify 3^(3x+3x) to 3^(6x): (9^(2x+1) - 3^(6x)) / 3^(3x) * 3^2
Step 8: Simplify expresseions with the same base
- Combine like terms by applying the properties of exponents:
- For any base a, a^m * a^n = a^(m+n)
- Simplify 3^(3x) * 3^2 to 3^(3x+2): (9^(2x+1) - 3^(6x)) / 3^(3x+2)
This is the simplified form of the given expression.