In Exercises 79–112, use rational exponents to simplify each expr...

Question

In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers.____⁹√x⁶y³

Accepted Answer

Welcome back. I am so glad you're here. We're asked to further simplify the following expression. Noting that any variable that appears in the expression represents positive numbers. The expression we're given is the 12th root of H raise to the eighth power K raise to the fourth power. Our answer choices are answer choice A, the cubed root of H squared K. Answer choice B, the cubed root of H K squared. Answer choice C the square root of two H K and answer choice DH square root K. All right, we recall from previous lessons that whenever we have an expression that is written with a radical, we can rewrite it with rational exponents, the rational exponents will be fractions. And so we can take this number here inside of the radical, starting with the H and that will be raised to the eight will be the numerator of that fraction. And the 12, the root of our radical will be the denominator of that fraction. The H is being multiplied by the K which will also be raised to a fraction. The K in our original expression is raised to the fourth and then four will be in the numerator for the K and that's divided by again 12, the R of the radical in the denominator, we can simplify both 8/12 and 4/12. All of these terms are divisible by the greatest common factor of four. So looking at 8/12, 8 divided by 4 is 2. And in the denominator, 12 divided by 4 is 3. So age is being raised to the 2/3. Now, for K 4/12 simplifies four, divided by four is one and 12, divided by four is three. So K is being raised to the one third. These two terms do not have like bases, but their exponents do have a common denominator. They're both three. So we can put this back into the form of a radical expression and we can take the three and that would be our new root for the radical. So we have the third, the cubed rut of and then remaining inside the radical, we have the H and that's raised to the numerator of its rational exponent. So it's raised to the second power H squared and the K is also raised to the num numerator of the rational exponent. So K raised to the first power, we can rewrite K raised to the first power is just K. So this simplifies as the cubed rut of H squared K. And that is as simplified as this one gets, we look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers.____⁹√x⁶y³

Key Concepts

Rational Exponents

Radical Notation

Simplifying Expressions

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