### Video Transcript

Which of the following is equal to negative one and a half cubed multiplied by negative one and a half squared? Is it A) negative seven and 19 over 32? B) Seven and 19 over 32? C) 32 over 243? D) Negative 3125 over 1024? Or E) negative 243?

As the numbers inside the brackets, or parentheses, are the same, we can use the law of indices, or exponents. 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏. We need to add the exponents, or indices. This means that negative one and a half cubed multiplied by negative one and a half squared is equal to negative one and a half to the power of five. As three plus two is equal to five.

Our next step is to convert the mixed number negative one and a half into a top-heavy or improper fraction. One whole one is equal to two-halves. Therefore, one and a half can be written as two-halves plus one-half. This is equal to three-halves or three over two. When the denominators are the same, we just add the numerators. We can, therefore, say that negative one and a half to the power of five is the same as negative three over two to the power of five.

We now need to use another one of the laws of indices, or exponents. Any fraction 𝑎 over 𝑏 raised to the power of 𝑥 can be rewritten as 𝑎 to the power of 𝑥 divided by 𝑏 to the power of 𝑥. This means that negative three over two or negative three-halves to the power of five is equal to negative three to the power of five divided by two to the power of five.

Negative three to the power of five can be written as negative three multiplied by negative three multiplied by negative three multiplied by negative three and multiplied by a fifth negative three. Two to the power of five can be rewritten in the same way.

A negative number multiplied by another negative number gives us a positive. Therefore, negative three multiplied by negative three is equal to positive nine. Grouping another two of the negative threes also gives us positive nine. We now need to multiply nine by nine by negative three. Nine multiplied by nine is equal to 81. And 81 multiplied by negative three is equal to negative 243.

Two to the power of five is equal to 32. Therefore, our denominator is 32. We can see at this point that our answer is negative. This immediately rules out option B and option C. It is also clear at this point that our answer is not the same as option D or E. Therefore, the correct option must be option A. We can check this by turning this improper fraction into a mixed number.

Our first step here is to divide negative 243 by 32. Six multiplied by 32 is equal to 192. Seven multiplied by 32 is equal to 224. And eight multiplied by 32 is equal to 256. This means that 243 divided by 32 is equal to seven with a remainder of 19. We can, therefore, rewrite negative 243 over 32 as negative seven and 19 over 32. Option A negative seven and 19 over 32 is equal to negative one and a half cubed multiplied by negative one and a half squared.