Video: Simplifying Rational Algebraic Expressions Using Laws of Exponents

Bethani Gasparine

Simplify (4^(3𝑛 + 3) × 25^(1 − 3𝑛))/(2^(9𝑛 + 3) × 50^(1 − 3𝑛)).

05:57

Video Transcript

Simplify four to the three 𝑛 plus three power times 25 to the one minus three 𝑛 power all divided by two to the nine 𝑛 plus three power times 50 to the one minus three 𝑛 power.

Here, we are working with bases of four, 25, two, and 50. And if we can get these bases to be alike these properties, 𝑥 to the 𝑎 times 𝑥 to the 𝑏 is equal to 𝑥 to the 𝑎 plus 𝑏 power and 𝑥 to the 𝑎 divided by 𝑥 to the 𝑏 is equal to 𝑥 to the 𝑎 minus 𝑏 power. And if we would have 𝑥 to the negative 𝑎 power, we could move that to the denominator and make the exponent positive. So it would be one over 𝑥 to the 𝑎.

So we need to somehow figure out how to rewrite these bases so they are all alike. So here are the bases that we’re working with. We can rewrite four as two squared. So it would be a base of two. Two is already a base of two. It’s two to the first power. 25 we can rewrite as five squared.

However, 50 isn’t a perfect square. But since we’ve been using bases of two and five, could we somehow rewrite 50 using them? Well, 50 is 25 times two and 25 is equal to five squared. So, we’ve written 50 using a base of five and two. So, let’s begin to simplify.

So beginning with our numerator, we will replace the four with two squared and 25 with five squared. And now to simplify, we must distribute the exponents. And we have two to the six 𝑛 plus six power times five to the two minus six 𝑛 power. We will now use this property, but backwards. So notice 𝑥 to the 𝑎 times 𝑥 to the 𝑏 equals 𝑥 to the 𝑎 plus 𝑏. And what we have so far? We have two to the six 𝑛 plus six power — so we need to separate those — and five to the two minus six 𝑛 power and we can separate those. So, we have two to the six 𝑛 plus six power.

So we could rewrite that as two to the six 𝑛 power times two to the six power and then five to the second power times five to the negative six 𝑛 power. And since we have a negative exponent, we will use this property. So we need to move this since it’s already on the numerator to the denominator and then we can change it to be a positive six 𝑛 power, which we would have here.

Now we also could have used the fact that when we subtract our exponents, it’s like dividing. So, we would have five to the second power divided by five to the six 𝑛 power. That’s how we have gotten the subtraction right here. Okay, so we’ve simplified the numerator. Now, let’s simplify the denominator.

So we’ll put a one on top to represent the fact that these numbers are on the denominator. So we have our bases of two and 50. So we will keep our base of two. And 50 we decided to rewrite it as five squared times two, which we’ve done here. So now, we need to simplify the five squared times two to the one minus three 𝑛 power. And now, we’ve separated them. Before moving on, let’s just distribute the two to the one and the negative three 𝑛. So we have five to the two minus six 𝑛 power.

Now, let’s use this property again to separate them. So we have first separated the nine 𝑛 and the three. So we’ve two to the nine 𝑛 power times two to the third power times five squared times five to the negative six 𝑛 power times two to the first power times two to the negative three 𝑛 power. So we’ve two negative powers. And we can make them positive by moving them up to the numerator. So now, we’ve completely simplified the denominator.

So here we had simplified our numerator. But we had moved five to the six 𝑛 to the denominator. And here, we simplified our denominator, but moved these up to the numerator. So we need to put them together. And we can put them together by multiplying because everything on top was supposed to go to the numerator and everything on the bottom was supposed to go to denominator.

Beginning with the numerator, we have two to the six 𝑛 and two to the three 𝑛 and we’re multiplying them. So, we can rewrite that as two to the six 𝑛 plus three 𝑛 power. And six 𝑛 plus three 𝑛 is nine 𝑛. Then, we had two to the six power times five to the second power times five to the six 𝑛 power. So, we didn’t put any of the fives together because we had a two and a six 𝑛. And those do not simplify together. We only put the two to the six 𝑛 and two to the three 𝑛 together because they were both 𝑛s.

On the denominator, we’ve five to the six 𝑛 times two to the nine 𝑛. And now, we have two to the third power and then two, just two to the first power. So, that would be two to the three plus one power, which is two to the fourth power, times five squared. So now a few things can cancel. The two to the nine 𝑛s cancel. The five squares cancel. The five to the six 𝑛 powers cancel. And we are left with two to the six power divided by two to the fourth power.

And when dividing like bases with exponents, we subtract their exponents and six minus four is two. And two squared is equal to four. So, after simplifying, our final answer will be four.

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