Algebra 2 Ration Exponents

Question

Megan and Julie are stuck simplifying radical expressions. Megan has to simplify the quantity of x to the one third power, over x to the one twelfth power. Julie has to simplify the thirty second root of the quantity of x times x to the second times x to the fifth. Using full sentences, describe how to fully simplify Megan and Julie's expressions. Describe if Megan and Julie started with equivalent expressions or if they started with expressions that are not equal.

Answer 1

Let us take Megan first. She has x^(1/3) divided by x^(1/12), and since the bases are the same, we use the quotient rule for exponents, which says you subtract the exponents:

x^(1/3) / x^(1/12) = x^(1/3 - 1/12)

Now we just have to subtract those fractions, which means finding a common denominator. Twelve works, since 1/3 is the same as 4/12:

4/12 - 1/12 = 3/12 = 1/4

So Megan's expression simplifies to x^(1/4), which is the fourth root of x.

Now Julie. She has the thirty second root of x times x^2 times x^5. Start inside the radical, because when you multiply powers with the same base you ADD the exponents, and remember that a lone x is really x^1:

x^1 times x^2 times x^5 = x^(1 + 2 + 5) = x^8

So she has the thirty second root of x^8. A root turns into a fractional exponent, where the index of the root becomes the denominator:

(x^8)^(1/32)

Now the power rule says to multiply the exponents:

x^(8/32)

And 8/32 reduces, since both divide by 8:

x^(1/4)

Remember, the last part of the question asks whether they started with equivalent expressions. Notice that both of them landed on exactly the same place, x^(1/4). So yes, Megan and Julie started with equivalent expressions. They just looked completely different because one was written as a quotient of fractional powers and the other was written as a root of a product. Once each one is simplified, they turn out to be two different disguises for the same thing.

So, our final answer is that both expressions simplify to x^(1/4), the fourth root of x, and the two girls did start with equivalent expressions.

Answer 2

Well, when you divide exponents with the same base, you sbtract the exponents.

 

For example, a³/a² = (a(a)(a))/(a(a)). 

 

a/a = 1, so it cancels out twice and leaves us with a/1 = a, which also equals a¹.

 

x^1/3/x^1/12 = x^{1/3 - 1/12} = x^{4/12 - 1/12} = x^3/12 = x^1/4

 

When you multiply, you add the exponents.

 

a³(a²) = a(a)(a)(a)(a) = a⁵

 

x^1/32(x²)(x⁷) = x^{9 1/32} = x^289/32, or the 32nd root of x raised to the 9th power.