Video: Simplifying Algebraic Expressions Using Laws of Exponents

Simplify (12π‘Žβ΅(11π‘ŽΒΉΒ³π‘ΒΉΒ³ βˆ’ 12π‘Žβ΅π‘ΒΉΒ³))/2π‘Žβ·π‘Β².

04:23

Video Transcript

Simplify 12π‘Ž to the fifth power times 11π‘Ž to the 13th power 𝑏 to the 13th power minus 12π‘Ž to the fifth power 𝑏 to the 13th power over two π‘Ž to the seventh power 𝑏 squared.

Remember, this fraction line means to divide. So essentially, here we’re dividing a polynomial by a monomial. Our first instinct might be to distribute the parentheses on our numerator. However, we can save ourselves a little bit of time. We’re going to begin by looking for the greatest common factor of 12π‘Ž to the fifth power and two π‘Ž to the seventh power 𝑏 squared.

Now, we’re allowed to do this and divide by this greatest common factor because we can see that any factor of 12π‘Ž to the fifth power must be a factor of the entire numerator. So let’s find the greatest common factor of these two terms. Now, we can do this by first considering the numerical parts, then considering the π‘Ž-part and then the 𝑏-part. The greatest common factor of 12 and two is two. Two is the largest number that divides evenly into 12 and two without leaving a remainder.

Then the greatest common factor of π‘Ž to the fifth power and π‘Ž to the seventh power is π‘Ž to the fifth power. There is no 𝑏 in 12π‘Ž to the fifth power. So we’ve found the greatest common factor. And before we do anything, we’re going to divide the numerator and the denominator of our fraction then by two π‘Ž to the fifth power.

To divide 12π‘Ž to the fifth power by two π‘Ž to the fifth power, we first divide 12 by two. That’s six. Then π‘Ž to the fifth power divided by π‘Ž to the fifth power is one. So 12π‘Ž to the fifth power divided by two π‘Ž to the fifth power is six. Similarly, we divide two π‘Ž to the seventh power 𝑏 squared by two π‘Ž to the fifth power. Two divided by two is one. Then, π‘Ž to the seventh power divided by π‘Ž to the fifth power is π‘Ž squared. 𝑏 squared divided by one is 𝑏 squared. So two π‘Ž to the seventh power 𝑏 squared divided by two π‘Ž to the fifth power is π‘Ž squared 𝑏 squared.

This means we can rewrite our fraction as six times 11π‘Ž to the 13th power 𝑏 to the 13th power minus 12π‘Ž to the fifth power 𝑏 to the 13th power over π‘Ž squared 𝑏 squared. Now, we’re ready to distribute our parentheses. We do so by multiplying six by the first term and then six by the second. Six times 11π‘Ž to the 13th power 𝑏 to the 13th power is 66π‘Ž to the 13th power 𝑏 to the 13th power. And then six times negative 12π‘Ž to the fifth power 𝑏 to the 13th power is negative 72π‘Ž to the fifth power 𝑏 to the 13th power. And this is all over π‘Ž squared 𝑏 squared.

Now, there are a number of ways we can evaluate the division at this point. Since that fraction line actually means divide, we could use the bus stop or we can reverse the process for adding fractions. And we can separate this into two individual fractions. Then, much as we did earlier, we’ll divide by the greatest common factor of our numerator and denominator. And if we wish, we can do this by the numerical part, then by the π‘Ž-part, then by the 𝑏-part separately.

Let’s make the denominator one π‘Ž squared 𝑏 squared. And then the greatest common factor of 66 and one is one. So we leave these for now. Then, what about the greatest common factor of π‘Ž to the 13th power and π‘Ž squared? Well, it’s π‘Ž squared. So we divide both the numerator and denominator by π‘Ž squared. This leaves π‘Ž to the 11th power on our numerator and then simply one on our denominator.

Similarly, the greatest common factor of 𝑏 squared and 𝑏 to the 13th power is 𝑏 squared. So when we divide through by 𝑏 squared, we’re simply left with 𝑏 to the 11th power on our numerator. And so our first term is 66π‘Ž to the 11th power 𝑏 to the 11th power over one, or simply 66π‘Ž to the 11th power 𝑏 to the 11th power.

Let’s repeat this process for our second fraction. 72 and one have a greatest common factor of one, so we leave them. Then, the greatest common factor of π‘Ž to the fifth power and π‘Ž squared is π‘Ž squared. Dividing through, and we’re left with π‘Ž cubed on the numerator. Then, the greatest common factor of 𝑏 squared and 𝑏 to the 13th power is 𝑏 squared. And as before, that leaves us with 𝑏 to the 11th power. And so we see that we’re left with 66π‘Ž to the 11th power 𝑏 to the 11th power minus 72π‘Ž cubed 𝑏 to the 11th power.

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