Video Transcript
Expand and simplify the expression negative two 𝑎 squared 𝑏 cubed 𝑐 times 𝑎𝑐 squared plus four 𝑎𝑏 minus 𝑏𝑐 squared.
In this question, we are given an algebraic expression, and we are asked to expand and simplify the expression. To do this, let’s start by looking at the given expression. We see that the expression includes the product in which one of the factors is a sum. This means that we will need to use the distributive property of multiplication over addition.
We can recall that the distributive property of multiplication over addition tells us that 𝑥 times 𝑦 plus 𝑧 is equals 𝑥𝑦 plus 𝑥𝑧. It is also worth noting that this property holds true no matter how many terms we are adding together. We can still distribute in the same way. We can also distribute over subtraction by subtracting the product instead of adding the product; 𝑥 times 𝑦 minus 𝑧 is 𝑥𝑦 minus 𝑥𝑧.
Distributing negative two 𝑎 squared 𝑏 cubed 𝑐 over the addition and subtraction gives us negative two 𝑎 squared 𝑏 cubed 𝑐𝑎𝑐 squared plus negative two 𝑎 squared 𝑏 cubed 𝑐 four 𝑎𝑏 minus negative two 𝑎 squared 𝑏 cubed 𝑐𝑏𝑐 squared.
Now that we have expanded the product, we need to simplify the resulting expression. To do this, we will need to use two properties. First, we can recall that the commutative property of multiplication tells us that we can reorder any product, so 𝑥𝑦 equals 𝑦𝑥. Second, we can recall that the product rule for exponents tells us that 𝑥 to the power of 𝑛 times 𝑥 to the power of 𝑚 is 𝑥 to the power of 𝑛 plus 𝑚.
We can apply these two results to each term in order to simplify. Let’s start with the first term. We can use the commutativity of multiplication to rewrite this term as negative two 𝑎 squared times 𝑎 times 𝑏 cubed 𝑐𝑐 squared. Next, we can recall that 𝑎 is the same as 𝑎 to the first power and 𝑐 is the same as 𝑐 to the first power. We can evaluate 𝑎 squared times 𝑎 and 𝑐 times 𝑐 squared by using the product rule for exponents. This gives 𝑎 to the power of two plus one and 𝑐 to the power of one plus two, respectively. This gives us negative two 𝑎 cubed 𝑏 cubed 𝑐 cubed.
It is worth noting that we are implicitly using the associativity property of multiplication to evaluate the product in any order. However, this is fine since we know that multiplication is associative.
We can follow this same process for the other two terms. For the second term, we can rearrange the product to obtain negative two times four multiplied by 𝑎 squared 𝑎𝑏 cubed 𝑏𝑐. We can simplify this expression by using the product rule for exponents. We get negative eight 𝑎 cubed 𝑏 to the fourth power 𝑐.
We now need to apply this process to the final term. We can rearrange the product using commutativity and then use the product rule for exponents to get negative two 𝑎 squared 𝑏 to the fourth power times 𝑐 cubed. Since we are subtracting this term, we can note that negative one times negative one is one to simplify further. This gives us negative two 𝑎 cubed 𝑏 cubed 𝑐 cubed minus eight 𝑎 cubed 𝑏 to the fourth power 𝑐 plus two 𝑎 squared 𝑏 to the fourth power 𝑐 cubed, which is our final answer.
