### Video Transcript

Expand two 𝑥𝑦 multiplied by five 𝑥 squared 𝑦 cubed minus two 𝑥 cubed 𝑦 squared.

In order to expand or multiply out the parentheses, we need to use the distributive property of multiplication. This involves multiplying two 𝑥𝑦 by five 𝑥 squared 𝑦 cubed and also multiplying two 𝑥𝑦 by two 𝑥 cubed 𝑦 squared.

In order to multiply the terms, we need to use one of the laws of indices: 𝑎 to the power of 𝑏 multiplied by 𝑎 to the power of 𝑐 is equal to 𝑎 to the power of 𝑏 plus 𝑐. If the base number is the same, we can add the indices or exponents.

If we firstly consider two 𝑥𝑦 multiplied by five 𝑥 squared 𝑦 cubed, we can see that two multiplied by five is equal to 10. Using the laws of indices, 𝑥 multiplied by 𝑥 squared is 𝑥 cubed or 𝑥 to the power of three. And in a similar way, 𝑦 multiplied by 𝑦 cubed is 𝑦 to the power of four.

We can add the exponents or the indices. One plus three equals four. Looking at the second part of our expansion, two multiplied by two is equal to four. 𝑥 or 𝑥 to the power of one multiplied by 𝑥 cubed is equal to 𝑥 to the power of four. And finally, 𝑦 multiplied by 𝑦 squared is 𝑦 cubed.

Therefore, the expansion of two 𝑥𝑦 multiplied by five 𝑥 squared 𝑦 cubed minus two 𝑥 cubed 𝑦 squared using the distributive property and the laws of indices is 10𝑥 cubed 𝑦 to the power of four minus four 𝑥 to the power of four 𝑦 cubed.