Exponential notation includes a base number and an exponent. The base number is the number that is being multiplied and the exponent is how many times the base number is being multiplied to itself.

In this case, \(\displaystyle 23\) is our base number and it's being multiplied to itself \(\displaystyle 6\) times, so that is our exponent.  

Apply the Power of a Product Principle:

\(\displaystyle (xy) ^{5} = x^{5} y ^{5}\)

Setting \(\displaystyle x^{5} = - 77\)  and \(\displaystyle y = -2\), keeping in mind that an odd power of a negative number is negative:

\(\displaystyle x^{5} y ^{5}\)

\(\displaystyle = -77 \cdot (-2 ) ^{5}\)

\(\displaystyle = -77 \cdot[ - (2 ^{5} )]\)

\(\displaystyle = -77 \cdot (-32)\)

\(\displaystyle = + (77 \cdot 32)\)

\(\displaystyle = 2,464\)

\(\displaystyle y^{2} = 12\) and \(\displaystyle y\) is positive, so 

\(\displaystyle y = \sqrt{12}\).

The greatest perfect square factor of 12 is 4, so the radical can be simplified:

\(\displaystyle y = \sqrt{4} \cdot \sqrt{3} = 2 \sqrt{3}\)

\(\displaystyle z^{2} = 5\), and \(\displaystyle z\) is positive, so \(\displaystyle z = \sqrt{5}\)

By the Power of a Power Property, 

\(\displaystyle (y z)^{3}\)

\(\displaystyle = y^{3} z^{3}\)

It is easiest to note that this can be broken up by the Product of Powers Principle, and evaluated by substitution:

\(\displaystyle y^{3} z^{3}\)

\(\displaystyle = y^{2+ 1} z^{2+ 1}\)

\(\displaystyle = y^{2 } \cdot y \cdot z^{2 } \cdot z\)

\(\displaystyle =12 \cdot \sqrt{12} \cdot 5 \cdot \sqrt{5}\)

\(\displaystyle =12 \cdot 5 \cdot \sqrt{12} \cdot \sqrt{5}\)

\(\displaystyle =60 \cdot \sqrt{60}\)

The greatest perfect square factor of 60 is 4, so the radical can be simplified:

\(\displaystyle 60 \cdot \sqrt{60}\)

\(\displaystyle =60 \cdot \sqrt{4} \cdot \sqrt{15}\)

\(\displaystyle =60 \cdot 2 \cdot \sqrt{15}\)

\(\displaystyle =120 \sqrt{15}\)

By the difference of squares pattern:

\(\displaystyle (4x +y) (4x -y)\)

\(\displaystyle = (4x) ^{2} - y^{2}\)

By the Power of a Power Principle,

\(\displaystyle (4x) ^{2} - y^{2}\)

\(\displaystyle = 4^{2}x^{2} - y^{2}\)

\(\displaystyle =16 x^{2} - y^{2}\)

Substituting 75 and 3 for \(\displaystyle x^{2}\) and \(\displaystyle y^{2}\), respectively:

\(\displaystyle =16 (75)- 3\)

\(\displaystyle =1,200- 3\)

\(\displaystyle =1,197\)

By the Power of a Power Principle, 

\(\displaystyle (y ^{5} ) ^{4} = y ^{5 \cdot 4} = y ^{20}\)

Substituting \(\displaystyle -7\) for \(\displaystyle y ^{5}\), keeping in mind that an even power of any number must be positive:

\(\displaystyle y ^{20} = (y ^{5} ) ^{4} =( -7)^{4} = 7 ^{4} = 7 \cdot 7 \cdot 7 \cdot 7= 2,401\)

By the perfect square trinomial pattern,

\(\displaystyle (x+ y) ^{2}\)

\(\displaystyle =x^{2} +2xy+ y ^{2}\)

\(\displaystyle x^{2} = 8\) and \(\displaystyle y^{2} = 18\).

Also, by the Power of a Power Principle, 

\(\displaystyle (xy) ^{2} = x^{2} y^{2} = 8 \cdot 18 = 144\), 

so, since \(\displaystyle x\) and \(\displaystyle y\) are both positive, 

\(\displaystyle xy = \sqrt{(xy) ^{2} }= \sqrt{144}= 12\).

Therefore, 

\(\displaystyle (x+ y) ^{2}\)

\(\displaystyle =x^{2} +2xy+ y ^{2}\)

\(\displaystyle = 8 +2 (12)+18\)

\(\displaystyle = 8 +24 +18\)

\(\displaystyle = 50\)

By the Power of a Power Principle, 

\(\displaystyle (y ^{3} ) ^{3} = y ^{3 \cdot 3} = y ^{9}\)

Therefore, we substitute, keeping in mind that an odd power of a negative number is also negative:

\(\displaystyle y ^{9} = (y ^{3} ) ^{3}\)

\(\displaystyle = (-17) ^{3}\)

\(\displaystyle = - 17 ^{3}\)

\(\displaystyle = - (17 \cdot 17\cdot 17)\)

\(\displaystyle = -4,913\)

By the perfect square trinomial pattern,

\(\displaystyle (x- y) ^{2}\)

\(\displaystyle =x^{2} -2xy+ y ^{2}\)

\(\displaystyle x^{2} =1 8\) and \(\displaystyle y^{2} = 8\).

Also, by the Power of a Power Principle, 

\(\displaystyle (xy) ^{2} = x^{2} y^{2} = 18 \cdot 8 = 144\), 

so, since \(\displaystyle x\) and \(\displaystyle y\) are both positive,

\(\displaystyle xy = \sqrt{(xy) ^{2} }= \sqrt{144}= 12\).

Therefore, 

\(\displaystyle (x- y) ^{2}\)

\(\displaystyle =x^{2} -2xy+ y ^{2}\)

And, substituting:

\(\displaystyle x^{2} -2xy+ y ^{2}\)

\(\displaystyle =18 -2 (12)+8\)

\(\displaystyle =18 -24 +8\)

\(\displaystyle = 2\)

Multiply out the expression by using multiple distributions and collecting like terms:

\(\displaystyle (t-x) (t^{2}+tx+x^{2})\)

\(\displaystyle = t (t^{2}+tx+x^{2}) - x (t^{2}+tx+x^{2})\)

\(\displaystyle = t \cdot t^{2}+ t \cdot tx+ t \cdot x^{2} - x \cdot t^{2}- x \cdot tx- x \cdot x^{2}\)

\(\displaystyle = t^{3}+ t^{2} x+ t x^{2} - t^{2} x- tx^{2} - x^{3}\)

\(\displaystyle = t^{3}+ t^{2} x- t^{2} x + t x^{2} - tx^{2} - x^{3}\)

\(\displaystyle = t^{3} - x^{3}\)

Since \(\displaystyle (t^{3} ) ^{2} = t ^{3 \cdot 2 } = t^{6}\) by the Power of a Power Principle,

\(\displaystyle t^{3} = \pm \sqrt{ t^{6}} = \pm \sqrt{144} = \pm 12\).

However, \(\displaystyle t\) is positive, so \(\displaystyle t^{3}\) is as well, so we choose \(\displaystyle t^{3} = 12\).

Similarly, 

\(\displaystyle x^{3} = \pm \sqrt{x^{6}} = \pm \sqrt{81} = \pm 9\).

However, since \(\displaystyle x\) is negative, as an odd power of a negative number, \(\displaystyle x^{3}\) is as well, so we choose \(\displaystyle x^{3} = -9\).

Therefore, substituting:

\(\displaystyle (t-x) (t^{2}+tx+x^{2}) = t^{3} - x^{3} = 12 - (-9) = 21\) 

If \(\displaystyle B\) is odd, then \(\displaystyle A + B +1\), the sum of three odd integers, is odd; an odd number taken to any positive integer power is odd.

If \(\displaystyle B\) is even, then \(\displaystyle A + B +1\), the sum of two odd integers and an even integer, is even; an even number taken to any positive integer power is even. 

Therefore, \(\displaystyle (A + B + 1)^{A+B+1}\) always assumes the same odd/even parity as \(\displaystyle B\).