Video: Manipulating Exponential Expressions

Factor out the highest common factor to rewrite the expression 2^(𝑦 + 3) − 8.

02:28

Video Transcript

Factor out the highest common factor to rewrite two to the power of 𝑦 plus three minus eight.

So the first step to rewriting our expression is to change eight into two raised to a power. And also we’ve demonstrated here eight is the same as two to the power of three because it’s two multiplied by two multiplied by two. So our expression now reads two to the power of 𝑦 plus three minus two to the power of three.

So now what I’m going to do is I’m gonna use an exponent rule to help us simplify it further. This exponent rule says 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. So you see, we’ve got the same base. And when we multiply with the same base, then we just add the exponents.

So therefore, what we can do is we can rewrite two to the power of 𝑦 plus three as two to the power of three multiplied by two to the power of 𝑦. And this is because we had 𝑦 plus three. So we used our exponent law to split it up. Okay, so now we’ve got two to the power of three multiplied by two to the power of 𝑦 minus two to the power of three.

So now, what we can do is factor. And we can factor by finding the highest common factor. Well, the highest common factor of both parts of our expression is two cubed or two to the power of three. And that’s because we’ve got two to the power of three multiplied by two to the power of 𝑦. And then, we’ve got two to the power of three. So therefore, we can have two to the power of three outside the parenthesis. And then, our first term within the parentheses is gonna be two to the power of 𝑦. And that’s because we had two to the power of three multiplied by two to the power of 𝑦.

And then, our second term is gonna be negative one. And that’s because if we multiply negative one by two to the power of three, we get negative two to the power of three, which is what we wanted. So therefore, we’ve got one more stage to simplify because we can rewrite two to the power of three as eight.

So therefore, we can say that if we factor out the highest common factor and rewrite two to the power of 𝑦 plus three minus eight, the result is eight multiplied by two to the power of 𝑦 minus one.

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