the function s(t)=(t^2+t)^2/3, t>0, represents the displacement s, in metres, of a particle moving along a straight line after t seconds

what a relief!

To find the displacement of the particle as a function of time, we can follow these steps:

Step 1: Start with the given equation: s(t) = (t^2 + t)^(2/3), where t represents time.

Step 2: Simplify the equation by raising the expression (t^2 + t) to the power of 2/3. To do this, we raise both the numerator (t^2 + t) and the denominator (3) to the power of 2:

s(t) = [(t^2 + t)^2]^(1/3) / 3^(1/3)

Step 3: Rewrite the equation using the exponent property that states (a^b)^c = a^(b * c):

s(t) = (t^2 + t)^(2/3) / 3^(1/3)

Step 4: Since the denominator 3^(1/3) is a constant, we can separate it from the squared term:

s(t) = (t^2 + t)^(2/3) * (3^(-1/3))

Step 5: Simplify the expression (3^(-1/3)). To do this, we take the cube root of 3, and then take its reciprocal:

3^(1/3) = 3^(1 * 1/3) = (3^(1))^(1/3) = 3^(1/3) = 1

Therefore, 3^(-1/3) = 1 / 3^(1/3) = 1 / 1 = 1.

Step 6: Substitute the simplified expression for (3^(-1/3)) back into the equation:

s(t) = (t^2 + t)^(2/3) * 1

Step 7: Finally, simplify the expression (t^2 + t)^(2/3). Raise both the numerator and denominator to the power of 2:

s(t) = [(t^2 + t)^2]^(1/3)

So, the final simplified equation for the displacement of the particle as a function of time is:

s(t) = (t^2 + t)^(2/3)

This equation represents the displacement (in meters) of a particle moving along a straight line after t seconds.