Question Video: Simplifying Rational Algebraic Expressions Using Laws of Exponents with Negative Exponents Mathematics

Video Transcript

Simplify 45 raised to the power 18𝑛 multiplied by 63 raised to the power negative nine 𝑛 multiplied by three raised to the power nine 𝑛 all divided by 225 raised to the power nine 𝑛 multiplied by 21 raised to the power negative nine 𝑛.

We’re given an exponential expression in the form of a fraction with various integer bases raised to powers which are multiples of 𝑛 in our numerator and denominator. To simplify this, we can use some of the laws or rules of exponents together with prime factorization of the bases. We begin by writing out our expression. Then we can use the negative exponent’s rule — that is, 𝑎 to the negative 𝑝 is one over 𝑎 to the power 𝑝, where 𝑎 is our base and 𝑝 our exponent — to simplify 63 raised to the negative nine 𝑛 and 21 raised to the power negative nine 𝑛. If our exponent is negative nine 𝑛, then 63 to the negative nine 𝑛 is one over 63 to the nine 𝑛. And similarly, 21 raised to the power negative nine 𝑛 is one over 21 to the nine 𝑛.

And recalling that to divide by a fraction, we flip and multiply. And to multiply by a fraction, we divide by the denominator so that 21 raised of the power nine 𝑛 is now in our numerator and 63 to the nine 𝑛 is in the denominator of our expression. So now that all of our exponents are positive, let’s break our integer bases down into their prime factorizations. Once we’ve done this, we’ll be able to collect terms with the same bases.

Our bases are 45, 21, 225, 63, and of course three. We know that 45 is nine times five, which is three times three times five. So that’s three squared times five. 21 is three multiplied by seven and doesn’t break down any further. 225 is three times three times five times five. That is three squared times five squared. 63 is nine times seven. That is three times three times seven, which is three squared times seven. And of course, the base three is its own prime factor.

So now we have three squared times five, that’s 45, raised to the power 18𝑛 multiplied by three times seven, that’s 21, raised to the power nine 𝑛 multiplied by three raised to the power nine 𝑛 in our numerator over three squared multiplied by five squared, that’s 225, raised to the power nine 𝑛 multiplied by three squared times seven, that’s 63, to the power nine 𝑛.

Now making some space, we see that in both our numerator and our denominator, we have products raised to a power, and we can use the power of a product rule to separate these out. The first term in our numerator, that’s three squared multiplied by five all raised to the power 18𝑛, becomes three squared to the power 18𝑛 multiplied by five raised to the power 18𝑛. The second term in our numerator, that’s three times seven to the power nine 𝑛, becomes three to the power nine 𝑛 multiplied by seven to the power nine 𝑛, which we multiply by our third term, three to the nine 𝑛. Similarly, in our denominator, we have three squared to the nine 𝑛 multiplied by five squared to the power nine 𝑛 multiplied by three squared to the nine 𝑛 multiplied by seven to the nine 𝑛.

At this point, we see we have a common factor in numerator and denominator of seven to the power nine 𝑛. So if we divide top and bottom by this, these become one. Next, we can apply the power law to the term with base five in our denominator. This gives us five raised to the power two raised to the power nine 𝑛 is five raised to the power two times nine 𝑛, which is five to the power 18𝑛. So now if we rewrite this in our denominator, we see that we have a common factor now of five raised to the power 18𝑛 in both numerator and denominator. And dividing both numerator and denominator by five raised to the power 18𝑛, these both become one. And now all of our bases are three.

So now if we introduce the product rule for exponents, that’s 𝑎 to the power 𝑝 multiplied by 𝑎 to the power 𝑞 is equal to 𝑎 to the power 𝑝 plus 𝑞, in our numerator, our two right-hand terms give us three raised to the power nine 𝑛 plus nine 𝑛, which is three to the power 18𝑛. And in our denominator, the same rule gives us three squared to the power nine 𝑛 plus nine 𝑛, which is three squared to the power 18𝑛. So now we have three squared to the power 18𝑛 multiplied by three to the power 18𝑛 divided by three squared to the power 18𝑛. So now if we divide both numerator and denominator by three squared raised to the power 18𝑛, these both become one, and we’re left with three raised to the power 18𝑛. The given expression can therefore be simplified to three raised to the power 18𝑛.