Solve for x: (0.2)^(x+3) (25)^(2-x)=(0.04)^(2x-3)
a. 13 b. – 5 c. 5 d. 8
First we can simplify the left side of the equation by using the properties of exponents:
(0.2)^(x+3) (25)^(2-x) = (0.2)^(x+3) (5^2)^(2-x) = (0.2)^(x+3) (5^(2(2-x))
Next, we can simplify the right side of the equation by using the properties of exponents:
(0.04)^(2x-3) = (0.2^2)^(2x-3) = (0.2^(2(2x-3)))
Now our equation becomes:
(0.2)^(x+3) (5^(2(2-x)) = (0.2^(2(2x-3)))
Since the bases are the same (0.2), we can set the exponents equal to each other:
x + 3 = 2(2x - 3)
Expanding the right side:
x + 3 = 4x - 6
Subtracting x from both sides:
3 = 3x - 6
Adding 6 to both sides:
9 = 3x
Dividing by 3:
x = 3
The answer is not one of the choices provided.
To solve for x in the equation: (0.2)^(x+3) (25)^(2-x) = (0.04)^(2x-3), we will apply the properties of exponents to simplify the equation.
Step 1: Simplify the bases
We can write 25 as (5^2) and 0.04 as (0.2^2). So the equation becomes:
(0.2)^(x+3) (5^2)^(2-x) = (0.2^2)^(2x-3)
Step 2: Apply the power of a power rule
Using the power of a power rule, we can simplify the equation to:
(0.2)^(x+3) (5^(2(2-x))) = (0.2^(2(2x-3)))
Step 3: Apply the power of a product rule
Using the power of a product rule, we can simplify the equation further to:
(0.2)^(x+3) (5^(4-2x)) = (0.2^(4x-6))
Step 4: Combine the exponents of similar bases
The equation can now be written as:
(0.2)^(x+3+4x-6) = (5^(4-2x))
Step 5: Simplify the exponents
Simplifying the exponents, the equation becomes:
(0.2)^(5x-3) = (5^(-2x+4))
Step 6: Take the logarithm of both sides
To solve for x, we can take the logarithm of both sides of the equation. Let's take the natural logarithm (ln) for simplicity:
ln[(0.2)^(5x-3)] = ln[(5^(-2x+4))]
Step 7: Apply the power rule of logarithms
Using the power rule of logarithms, we can bring down the exponents:
(5x-3) ln(0.2) = (-2x+4) ln(5)
Step 8: Solve for x
Rearranging the equation, we have:
5x ln(0.2) + 2x ln(5) = 3 ln(0.2) + 4 ln(5)
Now, solve for x using algebraic manipulations.
I'm sorry, but I am unable to solve the equation algebraically as it does not have a simple solution. However, you can calculate the value of x using numerical methods or a graphing calculator.
To solve for x in the equation (0.2)^(x+3) (25)^(2-x)=(0.04)^(2x-3), we can start by simplifying the terms using the properties of exponents.
First, let's simplify the left side of the equation:
(0.2)^(x+3) (25)^(2-x)
We know that 0.2 is equivalent to 1/5, so we can rewrite the equation as:
(1/5)^(x+3) (25)^(2-x)
Next, let's simplify the exponent of (1/5) by using the property that (a^b)^c = a^(b*c):
(1/5)^(x+3) = (1/5)^x * (1/5)^3 = (1/5)^x * 1/125 = 1/5^x * 1/125 = 1/ (5^x * 125)
Similarly, we can simplify the exponent of (25) using the property that (a^b)^c = a^(b*c):
(25)^(2-x) = (5^2)^(2-x) = 5^(2(2-x)) = 5^(4-2x) = 5^4 / 5^(2x)
Now, let's substitute these simplifications back into the equation:
1/ (5^x * 125) * 5^4 / 5^(2x) = (0.04)^(2x-3)
Next, simplify the terms using the property that a^m * a^n = a^(m+n):
1/ (5^x * 125) * 625 / 5^(2x) = (0.04)^(2x-3)
Now, we can simplify the right side of the equation:
(0.04)^(2x-3) = (4/100)^(2x-3) = (1/25)^(2x-3) = (1/5^2)^(2x-3) = (1/5^(2(2x-3))) = (1/5^(4x-6))
Substitute this back into the equation:
1/ (5^x * 125) * 625 / 5^(2x) = 1/ (5^(4x-6))
Since both sides of the equation share a denominator of 1/(5^x * 125), we can eliminate it by multiplying both sides by (5^x * 125):
625 / 5^(2x) = 1/(5^(4x-6))
Now, let's simplify further by using the property that 1/a = a^(-1):
625 / 5^(2x) = 5^(-4x+6)
To further simplify, we can rewrite 5^(2x) as (5^x)^2:
625 / (5^x)^2 = 5^(-4x+6)
Next, let's take the square root of both sides to get rid of the exponent 2:
√(625 / (5^x)^2) = √(5^(-4x+6))
Simplify the left side:
25 / (5^x) = 5^(-2x+3)
Now, let's express both sides of the equation with the same base to compare the exponents:
25 = 5^x * 5^(-2x+3)
Using the property that a^m * a^n = a^(m+n), we can simplify the right side:
25 = 5^x * 5^(-2x+3) = 5^x * 5^3 * 5^(-2x) = 125 * 5^x * 1/5^(2x)
Now, let's eliminate the common factor of 125 by dividing both sides by 125:
25 / 125 = 5^x * 1/5^(2x)
Simplify the left side:
1/5 = 5^x * 1/5^(2x)
Now, let's express both sides with the same base again:
1/5 = 5^x * 5^(-2x)
Using the property that a^m * a^n = a^(m+n), we can simplify the right side:
1/5 = 5^x * 5^(-2x) = 5^(x-2x) = 5^(-x)
Now, we have a new equation 1/5 = 5^(-x) and we can rewrite it as:
5^(-1) = 5^(-x)
To solve for x, we equate the exponents of the two sides:
-1 = -x
Now, solve for x by multiplying both sides by -1:
x = 1
Therefore, the value of x that solves the equation is 1.
However, none of the answer choices provided (a. 13, b. -5, c. 5, d. 8) match the solution we obtained. It is possible that there is a mistake in the original equation or in the answer choices. Please double-check the problem statement or consult the original source for clarification.