Solve for x, when (2^(x+1)^3)*4^x=8192.
Correction (2^(x+1)^3*4^x=8192. The whole expression, 2^(x+1) is raised to the power of 3.
To solve for x in the equation (2^(x+1)^3)*4^x = 8192, we'll start by simplifying the equation.
First, let's simplify the powers of 2 and 4:
2^(x+1)^3 = 2^(3(x+1)) = 2^(3x+3)
4^x = (2^2)^x = 2^(2x)
Now, substitute these simplified forms back into the equation:
2^(3x+3) * 2^(2x) = 8192
Next, we can use the property of exponents that states when two exponential expressions have the same base, we can multiply the exponents:
2^(3x+3+2x) = 8192
Combine like terms:
2^(5x+3) = 8192
To simplify further, we can express 8192 as a power of 2:
2^13 = 8192
Now, we can set the exponents equal to each other:
5x + 3 = 13
Subtract 3 from both sides:
5x = 10
Divide both sides by 5:
x = 2
Therefore, the solution to the equation is x = 2.
To solve for x in the equation (2^(x+1)^3)*4^x=8192, we need to simplify the expression and then isolate x.
Let's break down the equation step by step:
Step 1: Simplify the exponents inside the parentheses:
2^(x+1)^3 = (2^(x+1))^3
Step 2: Rewrite 4 as 2^2:
(2^(x+1))^3 * (2^2)^x = 8192
Step 3: Apply exponent rules for multiplication:
2^(3(x+1)) * 2^(2x) = 8192
Step 4: Apply the rule of adding the exponents with the same base:
2^(3x + 3) * 2^(2x) = 8192
Step 5: Combine the two terms with the same base into a single term:
2^(3x + 3 + 2x) = 8192
Step 6: Simplify the exponents:
2^(5x + 3) = 8192
Step 7: Write 8192 as a power of 2:
2^13 = 8192
Step 8: Set the exponents equal to each other:
5x + 3 = 13
Step 9: Solve for x:
5x = 10
x = 2
Therefore, the value of x that satisfies the equation is x = 2.