Answer :
To solve the equation
[tex]$$27^{x-6} = 9^{x-7},$$[/tex]
we start by expressing both bases in terms of [tex]$3$[/tex]:
[tex]$$27 = 3^3 \quad \text{and} \quad 9 = 3^2.$$[/tex]
Substitute these into the equation:
[tex]$$\left(3^3\right)^{x-6} = \left(3^2\right)^{x-7}.$$[/tex]
Using the power of a power rule, which states that [tex]$\left(a^b\right)^c = a^{bc}$[/tex], we simplify both sides:
[tex]$$3^{3(x-6)} = 3^{2(x-7)}.$$[/tex]
Since the bases on both sides of the equation are the same, we can equate the exponents:
[tex]$$3(x-6) = 2(x-7).$$[/tex]
Now, expand the expressions:
[tex]$$3x - 18 = 2x - 14.$$[/tex]
Subtract [tex]$2x$[/tex] from both sides:
[tex]$$3x - 2x - 18 = -14,$$[/tex]
which simplifies to:
[tex]$$x - 18 = -14.$$[/tex]
Add [tex]$18$[/tex] to both sides to solve for [tex]$x$[/tex]:
[tex]$$x = -14 + 18,$$[/tex]
so
[tex]$$x = 4.$$[/tex]
To verify, we can plug [tex]$x = 4$[/tex] back into the exponents:
- Compute the left exponent:
[tex]$$3(4-6) = 3(-2) = -6.$$[/tex]
- Compute the right exponent:
[tex]$$2(4-7) = 2(-3) = -6.$$[/tex]
Since both exponents are equal ([tex]$-6 = -6$[/tex]), the solution is verified.
Thus, the correct answer is [tex]$\boxed{4}$[/tex].
[tex]$$27^{x-6} = 9^{x-7},$$[/tex]
we start by expressing both bases in terms of [tex]$3$[/tex]:
[tex]$$27 = 3^3 \quad \text{and} \quad 9 = 3^2.$$[/tex]
Substitute these into the equation:
[tex]$$\left(3^3\right)^{x-6} = \left(3^2\right)^{x-7}.$$[/tex]
Using the power of a power rule, which states that [tex]$\left(a^b\right)^c = a^{bc}$[/tex], we simplify both sides:
[tex]$$3^{3(x-6)} = 3^{2(x-7)}.$$[/tex]
Since the bases on both sides of the equation are the same, we can equate the exponents:
[tex]$$3(x-6) = 2(x-7).$$[/tex]
Now, expand the expressions:
[tex]$$3x - 18 = 2x - 14.$$[/tex]
Subtract [tex]$2x$[/tex] from both sides:
[tex]$$3x - 2x - 18 = -14,$$[/tex]
which simplifies to:
[tex]$$x - 18 = -14.$$[/tex]
Add [tex]$18$[/tex] to both sides to solve for [tex]$x$[/tex]:
[tex]$$x = -14 + 18,$$[/tex]
so
[tex]$$x = 4.$$[/tex]
To verify, we can plug [tex]$x = 4$[/tex] back into the exponents:
- Compute the left exponent:
[tex]$$3(4-6) = 3(-2) = -6.$$[/tex]
- Compute the right exponent:
[tex]$$2(4-7) = 2(-3) = -6.$$[/tex]
Since both exponents are equal ([tex]$-6 = -6$[/tex]), the solution is verified.
Thus, the correct answer is [tex]$\boxed{4}$[/tex].
