Question Video: Simplifying an Expression Using the Product Rule of Exponents Mathematics

Fill in the blank 2π‘₯⁢ Γ— 6π‘₯Β² = οΌΏ.

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Video Transcript

Fill in the blank: two π‘₯ to the power of six times six π‘₯ squared is equal to blank.

In this question, we’re given an equation, and we’re asked to fill in the blank. We can see that the blank is the entire right-hand side of this equation. So it must be equal to the left-hand side of this equation. So to find the right-hand side of this equation, we’re going to need to simplify the left-hand side of this equation. And to do this, we need to notice something. Two π‘₯ to the power of six and six π‘₯ squared are both examples of monomials. Remember, a monomial is a product of constants and variables, where our variables are only raised to positive integer values, which is exactly what’s happening here because our exponents of π‘₯ are two and six, which are positive integers.

So to multiply these together, we need to recall our rule for multiplying monomials. We know π‘₯ to the power of π‘š multiplied by π‘₯ to the power of 𝑛 is equal to π‘₯ to the power of π‘š plus 𝑛. In other words, to multiply these together, we just add their exponents. The first thing we’re going to want to do is multiply our coefficients together. We want to multiply two by six. Of course, two multiplied by six is equal to 12. So our new coefficient is going to be 12.

Next, we need to multiply π‘₯ to the power of six by π‘₯ to the power of two. And we’re going to do this by adding the exponents together. π‘₯ to the power of six multiplied by π‘₯ to the power of two is π‘₯ to the power of six plus two. And of course, we know that six plus two is equal to eight. This gives us 12 times π‘₯ to the power of eight. And this is our final answer. Therefore, by using the product rule for monomials, we were able to show that two π‘₯ to the power of six times six π‘₯ squared is equal to 12π‘₯ to the power of eight.

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