In Exercises 85–116, simplify each exponential expression.(20a⁻³b...

Question

In Exercises 85–116, simplify each exponential expression.(20a⁻³b⁴c⁵/-2a⁻⁵b⁻²c)⁻²

Accepted Answer

Hello, today we're gonna be simplifying the given expression. So what we are given is 36 times R to the power of negative four times S to the power of six times T all over negative four times R to the power of negative 13 times S to the power of negative five times T to the power of seven, all of it race to the power of negative three. So before we evaluate the outermost exponent, let's go ahead and evaluate the insight, uh insight argument first. So let's group up the constants together, the R terms together the S terms together and the T terms together. So the constant of 36 over -4 is going to simplify to give us -9. And in order to simplify the rest of the variables, we can use the exponential property of division that states that we have A to the power of M over A, to the power of N, we can simplify the exponential values by taking the difference of the exponent and rewrite it as A to the power of M minus N. So for art to the power of negative four, divided by art to the power of negative 13, we can rewrite this as art to the power of negative four minus negative 13, which is going to be plus 13. And that is going to simplify to give us art to the power of nine. For the S terms, we have S to the power of six divided by S to the power of negative five. And that can be rewritten as S to the power of six minus negative five, which is going to give us six plus five. And that is going to simplify to give us S to the power of 11. And for the T terms, we have T divided by T to the power of seven, which can be written as T the power of one minus seven, which is going to simplify it to give us T to the power of negative six. So the expression can be rewritten as negative nine times R to the power of nine times S to the power of times T to the power of negative six all raised to the power of negative three. And now what we're going to do is we're going to distribute the exponent of negative three to each of the terms that is going to allow us to rewrite the expression As -9 race, the power of negative three times are raised to the power of nine race to the power of -3 times S to the power of 11 raised to the power of -3 times T race to the power of -6 race to the power of -3. And we can simplify the variables by using the property of negative exponents or exponential multiplication. And that states that if you have a to the power of M rate to the power of N, you can simplify this by taking the product of the exponents and rewrite this as a to the power of M times N. So for the R term, we're going to have our race of the power of nine times negative three, which is going to give us our race of the power of negative 27 S to the power of 11 times negative three is going to give us S to the power of negative 33 and T to the power of negative six times negative three is going to give us T to the power of 18. And for negative nine race to the power of negative three, what we're going to do is we're going to want to rewrite this exponent as a positive value. And to do that, we can use the property that states that if we have a to the power of negative M, we can take the reciprocal of this exponential value and turn the X1 in positive by rewriting this as one over A to the power of M. So if we apply this property, what we get is one over negative nine raised to the power of three and negative nine raced to the power of three is just negative nine times itself three times which is going to give us negative one over 729. So we have negative 1 over 729. Now notice that we have R reach to the power of a negative exponent and s reach to the power of the negative exponent. If we apply the inverse property of exponents, we can rewrite this to give us negative 100 -1 over times 1 over r to the power of 27 times one over S to the power of 33 times T to the power of 18. And if we multiply all of these values together, what we end up with is negative T to the power of 18, all over 729 times r to the power of 27 times s to the power of 33. And this is going to be the simplified form of the given expression. And with that being said, the answer to this problem is going to be D. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 85–116, simplify each exponential expression.(20a⁻³b⁴c⁵/-2a⁻⁵b⁻²c)⁻²

Key Concepts

Exponential Rules

Negative Exponents

Simplifying Rational Expressions

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