Question 1

Which of the following equations are true?

I.    (ab)⁰ = 1

II.   (a²b)³ = a⁵b³

III.  (a¹⁰b^-3)⁰ = z⁰

The following pictures are the exponent laws. You can use them as reference.

Select and check your answer...

A. I and II
B. I and III
C. I, II and III
D. All of the above

Let's take a look at:  I. (ab)⁰ = 1

We can treat ab as 'some number'. So, this term becomes:
                     (some number)⁰

Referring to the law as shown in the picture, we can see that (some number)⁰ is equals to 1.

Therefore, I. is true.

Let's take a look at:  II. (a²b)³ = a⁵b³

Oops! It seems that (a²b)³ doesn't fit into the exponent law as shown in the picture.

However, if we understand the logic behind the exponent laws, we can derive our own formula.

The following step will show you how.

Now, we know that:

              (a²b)³ = (a²b)(a²b)(a²b)
                       = a⁶b³

Now, realize that you can also get a⁶b³ by:
               
                     a⁶b³ = a^2x3 b^1x3    

With this, we can see that: (a²b¹)³ = a^2x3 b^1x3

Therefore, we can formulate: (aⁿb^m)^p = a^np b^mp

  *Note that b = b¹                           

Anyway, since we already found that: 

                    (a²b)³ = a⁶b³

The equation II. (a²b)³ = a⁵b³ is false.     

Let's take a look at:  III.  (a¹⁰b^-3)⁰ = z⁰

Again, we can treat a¹⁰b^-3 as 'some number'. So this term becomes:

                     (some number)⁰

Referring to the law as shown in the picture, we can see that (some number)⁰ is equals to 1.

Now, z⁰ is also equals to 1.  So, the equation becomes:    1 = 1

Therefore, III. is true.

From Step 1 to Step 5. The valid choices are I and III.

Clearly, the answer is B.