Could you help me on solving these please? thanks
-135 = -5b^-3
and
-9.6=-.3b^x
5/b^3 = 135
b^3 = 5/135 = 1/27
b = 1/3
Of course! I'd be happy to help you solve these equations.
Let's start with the first equation:
-135 = -5b^(-3)
To solve for b, we can start by isolating b on one side of the equation. To do this, we'll divide both sides of the equation by -5 first:
-135 / -5 = b^(-3)
This simplifies to:
27 = b^(-3)
Now, we need to get rid of the negative exponent. One way to do this is by taking the reciprocal of both sides:
1/27 = (b^(-3))^(-1)
This can also be written as:
1/27 = b^(3)
Now, we can rewrite the equation without the negative exponent:
b^(3) = 1/27
To solve for b, we need to take the cube root of both sides:
∛(b^(3)) = ∛(1/27)
The cube root of b^3 simplifies to b:
b = ∛(1/27)
Now, let's simplify the cube root. The cube root of 1 is 1, and the cube root of 27 is 3:
b = 1/3
Therefore, the solution to the equation -135 = -5b^(-3) is b = 1/3.
Now, let's move on to the second equation:
-9.6 = -0.3b^x
To solve for x, we'll start by isolating b^x on one side of the equation. To do this, divide both sides of the equation by -0.3:
-9.6 / -0.3 = b^x
This simplifies to:
32 = b^x
Now, we need to take the logarithm of both sides of the equation. Let's use the natural logarithm (ln) here:
ln(32) = ln(b^x)
Using the property of logarithms that states ln(a^b) = b * ln(a), we can rewrite the equation as:
ln(32) = x * ln(b)
Now, we can solve for x by dividing both sides of the equation by ln(b):
ln(32) / ln(b) = x
To find the numerical value of x, you will need to know the value of b, which is missing in the equation you provided. Without that information, we cannot calculate the exact value of x.
I hope this explanation helps you understand how to solve these equations. If you have any further questions, feel free to ask!