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.

(3y^¼)³
y^1/12

Accepted Answer

Welcome back. I am so glad you're here. We're asked to simplify the following expression by using the properties of rational exponents. Assuming that A is greater than zero, we are given the expression in the numerator. We have open parentheses. Four A raised to the 1/5 closed parentheses are raised to the fourth divided by, in the denominator. A raised to the 2/15. Our answer choices are answer choice. A one divided by 256. A raised to the 3/2. Answty Trace B 1 divided by 256. A race to the 2/3 answer choice C 256 a raised to the 3/2ves and answer choice. D 256. A raised to the two thirds. All right. So let's take a look at our numerator here. We have both four and a raise to the 1/5 being raised to the fourth power. So we need to distribute that four to both the four and the A raised to the 1/5 4 raised to the fourth power is 256 and A raised to the 1/5 raised to the fourth we recall from previous lessons that when we have a power raised to a power, we're going to multiply those exponents. So we have 1/5 multiplied by four, which equals 4/5. So we have a raised to the now 4/5 and that's still over a raised to the 2/15. We recall from previous lessons that when we have like bases, these are the a variables, they are the same base and they're being divided, we are going to subtract their exponents. So we are taking 4/5 minus 2/15. Well, in order to subtract fractions, they need to have a common denominator. The common denominator here would be 15. So we're going to multiply this first fraction, 4/5 by three, divided by three or three thirds because that simplifies to one. So now multiplying the numerators three, multiplied by four is 12 and that's divided by three, multiplied by five, which is 15. So we have 12 15th. Now we've got a common denominator, 12, 15th minus 2/15. We can subtract our numerators 12 minus two, that's 10 and our denominators remain 15. Now, we can simplify this. Both the numerator and the denominator are divisible by five. So taking 10 divided by five in the numerator, that's two and 15 divided by five and the denominator is three. So what we are left with for our A's exponent is two thirds. So we have 256. A raised to the two thirds. And this is as simplified as this one gets, we look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. (3y^¼)³ y^1/12

Key Concepts

Rational Exponents

Properties of Exponents

Simplifying Expressions

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