In Exercises 85–116, simplify each exponential expression.

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Question

In Exercises 85–116, simplify each exponential expression.

-24a³b⁻⁵c⁵/-3a⁻⁶b⁻⁴c⁻⁷

Accepted Answer

Hello everyone. We are gonna to simplify the following expression and write the answer with positive exponents. The expression we're given is open parentheses. Negative 70 P raised the fifth power Q raised to the negative 11th. Power R raised to the eighth. Power closed parentheses divided by open parentheses. Negative 14 P raised to the negative eighth. Power Q raised to the negative third. Power R raised the negative 14th power closed parentheses. Our answer choices are A 5 p. cubed Q raised the 14th power r raised the 6th power B five divided by open parentheses. P cubed Q raised the 14th power R raised the sixth power closed parentheses C open parentheses. Five P raised to the 13th. Power R raise the 22nd power closed parentheses divided by Q raised to the eighth power D five P raised to the 13th. Power Q raised to the eighth. Power R raised the 22nd power. First thing I'm gonna do is I'm gonna rewrite this division problem as a fraction. So my first set of parentheses will be my numerator. 2nd set my denominator. So rewriting this, I will have negative 70 P raised the fifth power Q raised the negative 11th power R raised the eighth power over negative 14 P raised the negative eighth power Q raised the negative third power R raise the negative 14th power. I like to rewrite it this way. So I can see all of my exponents, all of my variables a little clearer. So my next step, I'm gonna divide the coefficients. I have negative 70 divided by negative regardless of all of the variables and exponents, the coefficients divide like they have forever. So -70 divided by -14 is a positive five. That's gonna be in the numerator of our fraction. So I'm going to draw my fraction line. Next, I'm gonna move all of my negative exponents. I'm not gonna combine any of my variables yet. I just want to make everything be a positive exponent. Recall that when you move variables, the exponent if it's negative in the numerator is positive in the denominator and vice versa. So whichever part of the fraction it's in when it moves to the other part, it becomes positive. So looking at my numerator, I have P to the 5th power that's gonna stay put. I have Q raised the negative 11th power. So that's gonna move to the denominator as Q raised the 11th power R is the 8th power is gonna stay in the numerator. Looking at my denominator, I have P raise the negative eighth power. So I'm gonna move that to the numerator as P to the eighth power, Q raised the negative third power. So I'm moving that to the numerator as Q raised to the third power and R raise the negative 14th power. I'm gonna move that to the numerator is R raise the 14th power. So right now I have five P raise the fifth power, R raise the eighth power. P raise the eighth power. Q raised the third power. R raised the 14th power over Q raised the 11th power. Next, I'm gonna combine my like bases. And I recall that when I multiply variables with like bases, I add their exponents. So in my numerator, the five is staying put, I'm gonna write my fraction bar. I have P raise the fifth power and P raised the eighth power that will give me P raised the 13th power. Next, I have a Q raised to the third power. And I'm doing these in alphabetical order and then I have R raise the eighth power multiplied by R, raise the 14th power. That will be R, raise the 22nd power in my denominator. I still have qas the 11th power. So we're almost done this problem. We have one more set of variables to combine and that's our cues. I am actually gonna move Q cubed back to the denominator as Q raise the negative third power. And I'll show you why after I do it. So my numerator is now five P, raise the 13th power are raised to the 22nd power over Hugh raised the 11th power. He raised the negative 3rd power because now that I have both the cues in the denominator, I can multiply them together and add their exponents. So my numerator remains 5 p. raise the 13th power r raise the 22nd power and that's over when I add 11 and negative three, I get eight. So over eight ra or sorry over Q raised to the eighth power. So my final fraction is 5 p. raise the 13th power. R raise the 22nd power over Q raised the 8th power. All of our answer choices however, are written as division problems, not as fractions. And that's OK. Remember that every fraction we can lay it down and make it a division problem. So this is the numerator of five P. Raise the 13th power. R raise the 22nd power which I put in parentheses just to show it's all group together divided by Q raised the 8th power. This is our answer and matches answer choice. C have a nice day.

In Exercises 85–116, simplify each exponential expression. -24a³b⁻⁵c⁵/-3a⁻⁶b⁻⁴c⁻⁷

Key Concepts

Exponential Rules

Negative Exponents

Combining Like Terms

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