3^2x+1 -27^x=0 the x is 1 explain why and show working out
To solve the equation 3^(2x+1) - 27^x = 0, we can first simplify the equation by noting that 3^3 = 27.
So, 27 can be written as 3^3.
Now, we substitute 3^3 in place of 27 in the equation:
3^(2x+1) - (3^3)^x = 0
Next, we use the exponent rule that states when taking the power of a power, we multiply the exponents:
3^(2x+1) - 3^(3x) = 0
Now, we can continue solving the equation:
3^(2x+1) - 3^(3x) = 0
We can rewrite the equation in terms of a common base (3) by factoring out 3^2x:
3^(2x) * 3^1 - 3^(3x) = 0
Now, we notice that there is a common base of 3 in both terms:
3^(2x) * 3 - 3^(3x) = 0
Since we have a difference of squares, we can factor out a common factor of 3:
3 * (3^(2x) - 3^(3x)) = 0
To solve the equation, we set each factor equal to 0:
3 = 0 ---> This is impossible because 3 is not equal to 0.
3^(2x) - 3^(3x) = 0
Now, we can divide both sides of the equation by 3 to isolate the powers of 3:
(3^(2x) - 3^(3x))/3 = 0/3
Simplifying further:
3^(2x)/3 - 3^(3x)/3 = 0
Simplifying the left side by dividing powers of 3:
3^(2x - 1) - 3^(3x - 1) = 0
Now, to find the value of x that satisfies the equation, we can set the exponents equal to each other:
2x - 1 = 3x - 1
Simplifying the equation:
2x - 3x = 1 - 1
-x = 0
x = 0
Therefore, the value of x that satisfies the equation 3^(2x+1) - 27^x = 0 is x = 0.
To solve the equation 3^(2x+1) - 27^x = 0, follow these steps:
Step 1: Rewrite 27 as 3^3.
3^(2x+1) - (3^3)^x = 0
Step 2: Use the exponent properties to simplify the equation.
3^(2x+1) - 3^(3x) = 0
Step 3: Notice that both terms have a common base of 3, so we can apply the rule that states "if the bases are the same, then the exponents must be equal."
2x + 1 = 3x
Step 4: Rearrange the equation to isolate the variable.
2x - 3x = -1
-x = -1
Step 5: Multiply both sides by -1 to solve for x.
x = 1
Therefore, the value of x that satisfies the equation 3^(2x+1) - 27^x = 0 is x = 1.