Perform the indicated operation and simplify. Assume that all variables represent positive real

Perform the indicated operation and simplify. Assume that...

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and fifth roots. Understanding how to manipulate these expressions is fundamental in algebra, and it typically involves converting the radical form into exponent form, which simplifies the process of applying arithmetic operations.

Fractional Exponents

Fractional exponents provide a useful way to represent radical expressions; for example, the nth root of a variable can be expressed as raising that variable to the 1/n power. This representation leverages the rules of exponents, making it easier to multiply, divide, and simplify expressions that include radicals.

Properties of Exponents

The properties of exponents, such as the product and power rules, are essential tools for simplifying expressions with fractional exponents. These rules allow one to combine and simplify expressions by adding or multiplying exponents when the bases are alike.

Domain Restrictions in Radical Expressions

When working with radical expressions, particularly those involving even-index radicals like square roots, domain restrictions are crucial to ensure that the variables represent values for which the expressions are defined, typically positive real numbers.

Transcript

00:01 Alright, so in this question we have the cube root of x squared y times the square root of x y minus the fifth root of x y to the third.

00:23 So i'm going to make these have rational exponents, so this is x to the two thirds, y to the one third times x to the one half, y to the one half minus x to the one -fifth, y to the three -fits.

00:44 This is distributing, so we're going to distribute.

00:48 And so what we're going to get here then is we're going to get x to the two -thirds.

01:01 Actually, before we distribute, let's make them all have the same number, because that'll make our life a little easier.

01:07 And so we have threes and twos and fives, so that's going to be 30.

01:12 So this is going to be x to the 10 -30ths, y, to the 10, i'm sorry, that's going to be 2030th, y to the 1030th.

01:31 This would be x to the 1530th, y to the 1530th, minus x to the 630th, and y to the 1830th.

01:50 Now let's multiply, because now we can simply add those exponents together.

01:59 So when we start multiplying here, x the 20th 30th, that's going to be.

02:04 X then to the 35th 30th and y to the 25th 30th minus now we can do this this is going to be x to the 26th 30th and y to the 18 so y to the 28th 30th so we had all these 30th roots so this over here becomes the 30th root of x to the 35th y to the 25th minus the 30th root of x to the 26th and y to the 28th.

03:10 Now we do notice a lot of numbers that have them in common, and so they're probably we could have, let's go back up and we're going to try this, we're going to simplify...

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