In Exercises 59–72, simplify each expression using the products-t...

Question

In Exercises 59–72, simplify each expression using the products-to-powers rule.

(-2x⁻⁵y⁴z²)⁻⁴

Accepted Answer

Hey, everyone in this problem, we're asked to use the power rule to work out the given expression negative three X to the exponent negative seven Y cubed Z to the exponent four all to the exponent -5. We're given four answer choices. Option A negative X to the exponent multiplied by Y to the exponent 15 divided by 243 Z to the exponent 20. Option B negative X to the exponent divided by 243 Y to the exponent 15, Z to the exponent 20. Option C negative X to the exponent 35 Z to the exponent 20 divided by 243 Y to the exponent 15. And option D X to the exponent 35 divided by 243 Y to the exponent 15, Z to the exponent 20. We're gonna start by rewriting the expression we were given. So we have negative three X to the exponent -7. Why cubed Z to the exponent four and all of this to the exponent -5. Now, when we have a bunch of things multiplied together, all raised to an exponent that exponent needs to apply to each term. It means we can write this expression as negative three to the exponent -5 X to the exponent -7 to the exponent - Y cubed to the exponent - and Z to the exponent four to the exponent -5. OK. So that exponent of -5 is applied to every single one of these terms and they're multiplied together. Now let's simplify the exponents on our X Y and Z terms. We're called the power rule. We have a to the exponent M to the exponent N, we can write this as a to the exponent M multiplied by N. So if we have an exponent raised to an exponent, we can simplify and combine these exponents together by multiplying them. If we apply this to our problem, we're gonna leave the negative three to the exponent negative five alone. For this step, we have X to the exponent negative seven to the exponent negative five. We multiply the exponents and we get X to the exponent 35 doing the same for the Y term Y cubed to the exponent negative five. We multiply the exponents, we get Y to the exponent negative 15, Z to the exponent 4 to the exponent -5. We multiply the exponents and we get Z to the exponent -20. Now we still need to work out this coefficient in front. You also notice our answer choices all have positive exponents. So our Y term and our Z term have negative exponents. We need to write those as positive exponents. We recall if we have a to the exponent negative M and negative exponent, we can write this as one divided by A to the exponent M. So we can put the base and the exponent in the denominator and that exponent becomes positive. Applying this to our problem, we have negative three to the exponent negative five. By applying the rule, this is equivalent to one divided by -3 to the exponent five. We multiply by X to the exponent Y to the exponent -15. You can write this as one divided by Y to the exponent 15. Similarly for Z to the exponent -20, you can write this as one divided by Z to the exponent 20. OK. So we move those into the denominator. The exponent becomes positive. Now we're getting close, let's deal with this coefficient. We have one divided by -3 to the exponent five. OK. Negative three to the exponent five gives us negative 243. And so we can write this expression as negative X to the exponent 35 divided by 243. OK. We got negative 243 and we've just written the negative in the numerator so that it corresponds with our answer choices Y to the 15 Z to the 20 in that denominator as well. And this is the simplified expression, we've written it with all positive exponents to match with the answer choices. And we found that the correct answer is negative X to the exponent 35 divided by 243 Y to the exponent 15, Z to the exponent 20 which corresponds with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 59–72, simplify each expression using the products-to-powers rule. (-2x⁻⁵y⁴z²)⁻⁴

Key Concepts

Products-to-Powers Rule

Negative Exponents

Simplifying Expressions

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