Question

Simplify each expression. Assume that all variables are positive when they appear. $$8 x y-\sqrt{25 x^{2} y^{2}}+\sqrt[3]{8 x^{3} y^{3}}$$

Short answer

5xy

Step-by-step solution

- Simplify each term separately

Identify each term and simplify them individually. The expression is: 1. The first term is \(8xy\). 2. The second term is \(\sqrt{25x^2y^2}\). 3. The third term is \(\sqrt[3]{8x^3y^3}\).

- Simplify the second term

Simplify \(\sqrt{25x^2y^2}\): \[\sqrt{25x^2y^2} = \sqrt{25} \cdot \sqrt{x^2} \cdot \sqrt{y^2}\] \[= 5xy\]

- Simplify the third term

Simplify \(\sqrt[3]{8x^3y^3}\): \[\sqrt[3]{8x^3y^3} = \sqrt[3]{8} \cdot \sqrt[3]{x^3} \cdot \sqrt[3]{y^3}\] \[= 2xy\]

- Combine all simplified terms

Combine the simplified terms: \(8xy - 5xy + 2xy\) \[8xy - 5xy + 2xy = (8 - 5 + 2)xy = 5xy\]

Key concepts

Radicals

Radicals are expressions that include a root, like a square root or a cube root. Simplifying radicals means breaking them down into their simplest form.
For example, in the given exercise, we need to simplify \(\frac{25}{x^2 y^2}\).

The square root of a number is a value that, when multiplied by itself, gives the original number. So, \(\frac{\backslash sqrt{25} = 5\) because 5 * 5 = 25.
Also, \(\frac{\backslash sqrt{x^2} = x\) because x * x = x^2, and \(\frac{\backslash sqrt{y^2} = y\) because y * y = y^2.
The square root of 25x^2y^2 is \(\frac{5xy}\).Combining all of these, we get:
\(\frac{\backslash sqrt{25x^2 y^2} = 5xy\)
Understanding how to work with radicals is crucial for simplifying expressions.

Cubic Roots

Cubic roots are another type of root; they involve finding a value that, when used in multiplication three times, gives the original number.
Let's look at the example \(\frac{backslash sqrt[3]{8x^3 y^3}}\).
The cubic root of 8 is 2, because 2 * 2 * 2 = 8.
Similarly, the cubic root of \(x^3\) is x, because x * x * x = x^3.
And the cubic root of \(y^3\) is y, because y * y * y = y^3.
Putting these together, we get:
\(\frac{\backslash sqrt[3]{8x^3 y^3} = 2xy\).
Breaking down and understanding cubic roots is important to handle and simplify these expressions.

Combining Like Terms

Combining like terms is a fundamental algebraic process. It involves simplifying expressions by adding or subtracting terms that have the same variable parts.
From our exercise, once we simplified \(8xy - 5xy + 2xy\), we combine like terms:
\(\frac{8xy}\) and \(5xy\) and \(2xy\) are like terms because they have the same variables.
So we perform the arithmetic on the coefficients and keep the variable parts the same:
\(8xy - 5xy + 2xy = \backslash (8 - 5 + 2\backslash)xy = 5xy\).
This step-by-step process makes the expression easier to handle and understand.