Simplify
(3a^-2b^3)^-4
To simplify the expression, we can apply the exponent rules.
First, we can distribute the exponent of -4 to both the numerator and denominator of the expression.
Doing so, we get:
((3^-4) * (a^-2*-4) * (b^3*-4))
Next, we can simplify the exponents:
((1/3^4) * (a^8) * (b^-12))
Simplifying further, we get:
(a^8) / (81 * b^12)
A jet takes 8hours to travel a distance of 7168 miles against the wind, the. Return trip take 7 hours with the wind. What is the rate if the jet in still air and what is the rate of the wind
Let's denote the rate of the jet in still air as "x" and the rate of the wind as "y".
Against the wind:
Rate = x - y
Time = 8 hours
Distance = 7168 miles
Using the formula Distance = Rate * Time, we can write the equation:
7168 = (x - y) * 8
With the wind:
Rate = x + y
Time = 7 hours
Distance = 7168 miles
Using the same formula, we can write the equation:
7168 = (x + y) * 7
We now have a system of equations:
8(x - y) = 7168
7(x + y) = 7168
Expanding and simplifying these equations, we get:
8x - 8y = 7168
7x + 7y = 7168
To solve the system of equations, we can multiply the second equation by 8 and subtract it from the first equation:
8x - 8y - (56x + 56y) = 7168 - (57264)
-48x = -50096
Dividing both sides by -48, we get:
x = 1046
Substituting this value into the first equation, we can find the value of y:
8x - 8y = 7168
8(1046) - 8y = 7168
8368 - 8y = 7168
-8y = -1200
Dividing both sides by -8, we get:
y = 150
Therefore, the rate of the jet in still air is 1046 mph and the rate of the wind is 150 mph.
Write and equation for the expression p varies directly with f and inversely with u . Use k as the constant of proportionality
The equation for the expression "p varies directly with f and inversely with u" can be written as:
p = k * (f/u)
In this equation, p represents the variable that varies directly with f and inversely with u. k is the constant of proportionality. Multiplying k by the ratio of f to u represents the direct variation with f and the inverse variation with u.
To simplify the expression (3a^-2b^3)^-4, we first need to expand the parentheses and then apply the exponent to each term inside.
Inside the parentheses, we have 3, a^-2, and b^3.
Let's start by expanding the expression inside the parentheses:
(3a^-2b^3)^-4 = 3^-4 * (a^-2)^-4 * (b^3)^-4
When a negative exponent is raised to a power, the negative exponent becomes positive.
So, (a^-2)^-4 becomes a^8, and (b^3)^-4 becomes b^-12.
Now our expression becomes:
3^-4 * a^8 * b^-12
Next, let's simplify the expression further.
Since 3 is a constant, 3^-4 can be rewritten as 1/3^4 or 1/81.
So, the simplified expression is:
1/81 * a^8 * b^-12
We can also simplify further by combining the terms with the same base.
Since a^8 and b^-12 have different bases, they cannot be combined.
Therefore, the simplified expression is:
1/81 * a^8 * b^-12
To simplify the expression (3a^(-2)b^3)^(-4), we need to apply the laws of exponents.
Step 1: Distribute the exponent -4 to the terms inside the parentheses.
Recall that when you have a negative exponent, you can take the reciprocal of the base and make the exponent positive. In this case, we will apply the negative exponent to each term inside the parentheses.
(3^(-4) * a^((-2)(-4)) * b^(3(-4)))
Step 2: Simplify each term inside the parentheses.
First, simplify the constants 3^(-4) = 1/3^(4) = 1/81.
Next, we apply the power of a power rule for exponents (a^m)^n = a^(m*n) to the variables.
a^((-2)(-4)) = a^(8). Since -2 * -4 = 8.
b^(3(-4)) = b^(-12). Since 3 * -4 = -12.
Therefore, the expression simplifies to:
1/81 * a^8 * b^(-12)
To summarize, (3a^(-2)b^3)^(-4) simplifies to 1/81 * a^8 * b^(-12).