Given circle p with arc ae = 53 , arc ba =68 and arc cb =72

match the following angles with their corresponding measurements

The theory that is mainly used to find out the angles of the above diagram is that;
the sum of the internal angles of a triangle is 180°.
when the angles of arcs have been given(e.g: BA) that means from the centre of the circle to point B has been rotated by 53° to point A. 
Next from point A to point E, the angle rotated is 68°

the addition of these 2 angles give the angle rotated from point B to point E,
that is angle no.7

m∠7 = 53° + 68° = 121°
consider ΔPBE,
PB and PE are radii therefore isosceles triangles
In isosceles triangles the angles opposite equal sides are equal 
if the angle is x°, then there are 2 angles with values x each 
x° +x° + 121° = 180°
2x° = 59°
x = 29.5°
therefore m∠3 = 29.5°

next connect PC and consider ΔPCB,
according to given information, C∠PB = 72°
as PC and PB are radii ΔPCB is an isosceles triangle, then the angles (y°) opposite radii are equal 
2y° + 72° = 180°
y° = 54°
then consider ΔPCD, again an isosceles triangle 
as a straight line has an angle of 180°
B∠PC + C∠PD = 180°
72° + C∠PD = 180°
C∠PD = 108°
as ΔPCD is an isosceles triangle, P∠CD and P∠DC are equal (take this angle as z°)
2z° + 108° = 180°
z° = 36°
B∠CD is a sum of y and z angles
therefore m∠8 = y + z 
m∠8 = 54° + 36° = 90°
there is also a tangent line that cuts point D of the circle. this tangent cuts D and is perpendicular to the PD radius
this indicates that lines PD and HD are perpendicular to each other
hence, forms a right angle where it cuts each other.
therefore H∠DP / m∠4 = 90°
C∠DH /m∠5 is the difference between P∠DH and P∠DC
m∠5 = 90° - 36° = 54°

m∠1 and m∠6 together make a straight line therefore the sum of these 2 angles is 180°
by looking at the options of answers given the 2 angle values given that add up to 180° are 116.5° and 63.5°.
m∠1 is the acute angle therefore m∠1 = 63.5°
m∠6 is the obtuse angle therefore m∠6 = 116.5°

consider ΔDFB
D∠FB is 180° - m∠1 = 180° - °63.5 = 116.5°
D∠FB + F∠DB(w°) + D∠BF (m∠3)= 180°
116.5 + w° + 29.5° = 180°
w° = 34°
m∠2 = m∠4 + w°
        = 90° + 34° 
       = 124°

finally all the answers are 
m∠1= 63.5°
m∠2 = 124°
m∠3 = 29.5°
m∠4 = 90°
m∠5 = 54°
m∠6 = 116.5°
m∠7 = 121°
m∠8  = 90°

Hello!

the answer is:

the answer is:

why?

to solve the problem, we need to remember the quotient of power property, it's defined by the following relation:

if we have a quotienf of powers that have the same base, we need to keep the same base and subtract the exponent of the denominator power to the exponent of the numerator power.

so, we are given the expression:

then, calculating we have:

hence, the answer is:

have a nice day!