Question

\(12 \sqrt[3]{y}+9 \sqrt[3]{y}\)

Short answer

The simplified form of the expression \(12 \sqrt[3]{y}+9 \sqrt[3]{y}\) is \(21 \sqrt[3]{y}\).

Step-by-step solution

Recognise like terms

The terms \(12 \sqrt[3]{y}\) and \(9 \sqrt[3]{y}\) are similar, since they both contain the cube root of \(y\). Similar or like terms are those terms which have the same variable and the same exponent. Here both terms have the cube root of \(y\) which makes them similar.

Addition of cube roots

When adding like terms, just combine the coefficients of the terms. This gives us: \(12 \sqrt[3]{y} + 9 \sqrt[3]{y} = (12 + 9) \sqrt[3]{y}\).

Simplify the sum

Calculate the sum of 12 and 9 to get: \((12 + 9) \sqrt[3]{y} = 21 \sqrt[3]{y}\).

Key concepts

Cube Roots

A cube root is a number that, when multiplied by itself three times, gives the original number. It is represented as \( \sqrt[3]{x} \) where \( x \) is the number or expression under the cube root. For instance, the cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\). Similarly, in the expression \( \sqrt[3]{y} \), we are looking for a number which when cubed equals \( y \). Cube roots are essential when dealing with algebraic expressions, particularly when analyzing the roots of a polynomial or solving equations that involve cubed terms.
Cube roots can appear in various forms, including:

• Numerical cube roots, like \( \sqrt[3]{8} = 2 \).
• Algebraic cube roots, like \( \sqrt[3]{y^3} = y \).

Understanding cube roots is crucial to simplifying expressions and solving equations, especially those involving polynomials and radicals.

Like Terms

In mathematics, like terms are terms that contain the same variables raised to the same power. They make it easier to simplify algebraic expressions. For example, in the expression \( 12 \sqrt[3]{y} + 9 \sqrt[3]{y} \), both terms are considered like terms because they involve the cube root of \( y \).
Identifying like terms is important because:

• They can be combined through addition or subtraction, simplifying the expression.
• Only like terms can be combined to create a simplified expression.

To combine like terms, focus on the coefficients and ensure the variable parts are identical. This allows us to consolidate them into a single term, making complex expressions easier to work with.

Coefficients Addition

Coefficients are numerical factors in terms of algebraic expressions. When dealing with like terms, as in our example \( 12 \sqrt[3]{y} + 9 \sqrt[3]{y} \), the coefficients are 12 and 9. To simplify the expression, we add the coefficients while keeping the like terms' variable parts the same.
The addition of coefficients simplifies the expression by reducing it to fewer terms:

• Add the coefficients: \( 12 + 9 = 21 \).
• Retain the common variable part of the terms, here \( \sqrt[3]{y} \).

This leads to the final, simplified expression \( 21 \sqrt[3]{y} \). Understanding how to efficiently add coefficients allows for quick simplification of expressions with like terms, which is a fundamental skill in algebra.