In Exercises 55–78, use properties of rational exponents to simpl...

Question

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers.

(2x^⅕)⁵

Accepted Answer

Hey, everyone in this problem, we're asked to simplify the following expression by using the properties of rational exponents. We're told to assume that M is greater than zero. The expression we're given is four M to the exponent 1 4th all to the exponent four. We're given four answer choices. Option A 256 M to the exponent 17 divided by four. Option B four M option C 256 M and option D four M to the exponent 17 divided by four. We're gonna start by rewriting what we were given. So in brackets, we have four M to the exponent 1/4 and all of this is raised to the exponent four. Now we have two things multiplied together that are raised to this exponent four. We have four multiplied by M to the exponent 1/4. OK. So this exponent four needs to apply and distribute to both of those terms. How do we do that? I recall if we have two things multiplied together, say X to the or sorry, X multiplied by Y raise to an exponent N. This is equivalent to writing X to the exponent N multiplied by Y to the exponent N. OK. So that exponent N applies to both terms. We can split them up, do the exponent to on each term and then multiply the result together. So applying this to our case, we're gonna have four to the exponent four multiplied by M to the exponent 1/ to the exponent four. Now we can deal with each of these terms separately multiplying the result together, which is gonna be easier than trying to figure out the entire result altogether. So we have four to the exponent four. It's gonna give us 256. We have M to the exponent 1/4 to the exponent four. How do we simplify an exponent of an exponent? We'll recall the power rule. We have something, say X to the exponent A to the exponent B. We can simplify this exponent of an exponent and write it as a single exponent where we multiply the two original ones together. So this is equal to X to the exponent A multiplied by B. If we apply that to our problem, we have to the exponent 1/4 to the exponent four. We're gonna multiply the exponents together 1/4 multiplied by four, gives us one. So we go M to the exponent one. And so we can write this as just 256 M and we don't need to write that exponent of one that's implied. Now, there's nothing else to simplify. This is as simple as we can get this expression. And that's the result we were looking for. We found the correct answer is 256 M which corresponds with answer choice. C Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. (2x^⅕)⁵

Key Concepts

Rational Exponents

Properties of Exponents

Simplification of Expressions

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