### Video Transcript

Complete four 𝑥 squared 𝑦 cubed multiplied by something plus something minus two 𝑥 cubed is equal to negative 12𝑥 to the power of four 𝑦 to the power of six plus 12𝑥 squared 𝑦 to the power of six plus something.

In order to answer this question, we need to find the missing terms 𝑎, 𝑏, and 𝑐. In order to expand or multiply out a bracket, we need to use the distributive property. This involves multiplying everything outside the bracket, in this case four 𝑥 squared 𝑦 cubed, by everything inside the bracket. This means that multiplying four 𝑥 squared 𝑦 cubed by 𝑎 gives us negative 12𝑥 to the power of four 𝑦 to the power of six. If we divide both sides of the equation by four 𝑥 squared 𝑦 cubed, we are left with 𝑎 is equal to negative 12𝑥 to the power of four 𝑦 to the power of six divided by four 𝑥 squared 𝑦 cubed.

At this point, we need to consider one of our laws of indices. 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus 𝑏. We can subtract the exponents or indices. Going back to our equation, negative 12 divided by four is negative three. 𝑥 to the power of four divided by 𝑥 squared is 𝑥 squared, as we can subtract the exponents. And finally, 𝑦 to the power of six divided by 𝑦 cubed is 𝑦 cubed. This means that the first missing value in the equation is negative three 𝑥 squared 𝑦 cubed.

We can use the same method to work out 𝑏. Four 𝑥 squared 𝑦 cubed multiplied by 𝑏 is equal to 12𝑥 squared 𝑦 to the power of six. Once again, we can divide by four 𝑥 squared 𝑦 cubed. This gives us 𝑏 is equal to 12𝑥 squared 𝑦 to the power of six divided by four 𝑥 squared 𝑦 cubed. Using our laws of indices once again, this gives us an answer for 𝑏 equal to three 𝑦 cubed as 12 divided by four is three. 𝑥 squared divided by 𝑥 squared is one. And 𝑦 to the power of six divided by 𝑦 cubed is 𝑦 cubed. This means that the second missing value in our equation is three 𝑦 cubed.

The third missing term in our question is on the right-hand side of the equal sign. This means that using the distributive property, four 𝑥 squared 𝑦 cubed multiplied by negative two 𝑥 cubed is equal to 𝑐. In order to simplify this, we need to use a different law. 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. In this case, this gives us an answer for 𝑐 of negative eight 𝑥 to the power of five 𝑥 cubed, as four multiplied by negative two is negative eight and 𝑥 squared multiplied by 𝑥 cubed is 𝑥 to the power of five. This means that our third and final missing term in the equation is: 𝑐 is equal to negative eight 𝑥 to the power of five 𝑦 cubed.

We could substitute these terms back into the initial equation and expand the left-hand side to check that our answer is correct.