### Video Transcript

Which of the following is equal to two and a half cubed multiplied by two and a half squared? Is it (A) 3125 over 32, (B) 15625 over 64, (C) 75 over two, (D) two and one over 32, or (E) two and a half squared all to the power of five?

As the fractions inside the brackets are equal, one way to start this problem would be to use one of the laws of exponents or indices. This states that 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. When multiplying a number raised to two exponents, we can add the exponents or indices. In this question, two and a half cubed multiplied by two and a half squared is equal to two and a half to the power of five. This is because three plus two is equal to five.

Our next step is to convert the mixed number two and a half into an improper or top-heavy fraction. Two is the same as four-halves or four over two. This means that two and a half is equal to four-halves plus one-half. This is equal to five-halves. So two and a half can be rewritten as five over two. A quicker way to work out the numerator of this improper fraction is to multiply the whole number, two, by the denominator, two, and then add the numerator, one. Two multiplied by two is equal to four. Adding one gives us five.

Going back to our question, two and a half to the power of five can be rewritten as five over two to the power of five. Another one of our laws of indices or exponents states that any fraction 𝑎 over 𝑏 that is 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 five over two to the power of five can be rewritten as five to the power of five divided by two to the power of five. Five to the power of five is equal to 3125 as five multiplied by five multiplied by five multiplied by five multiplied by five is equal to 3125. In the same way, two to the power of five is equal to 32. We can therefore say that option A, 3125 over 32, is equal to two and a half cubed multiplied by two and a half squared.