Question

Simplify each expression. Assume that all variables are positive when they appear. $$3 x \sqrt{9 v}+4 \sqrt{25 v}$$

Short answer

The simplified expression is \( 9x \sqrt{v} + 20 \sqrt{v} \).

Step-by-step solution

Identify the square roots

Find the square roots inside the expression. The terms are \ \( \sqrt{9v} \) and \( \sqrt{25v} \).

Simplify each square root

For each square root, find the value of the square root and simplify. \ \[ \sqrt{9v} = \sqrt{9} \cdot \sqrt{v} = 3 \sqrt{v} \] \ \[ \sqrt{25v} = \sqrt{25} \cdot \sqrt{v} = 5 \sqrt{v} \]

Combine like terms

Multiply the coefficients outside the square roots by the simplified square roots. \ \[ 3x \sqrt{9v} = 3x \cdot 3 \sqrt{v} = 9x \sqrt{v} \] \ \[ 4 \sqrt{25v} = 4 \cdot 5 \sqrt{v} = 20 \sqrt{v} \]

Add and simplify the terms

Add the resulting expressions, combining like terms. \ \[ 9x \sqrt{v} + 20 \sqrt{v} \] \ The terms cannot be combined further due to different coefficients of \( \sqrt{v} \).

Key concepts

Square Roots

Square roots are fundamental in simplifying expressions involving radicands. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because \(3 \times 3 = 9\). When expressions include variables, the square root process is similar. Consider:

• The term \( \sqrt{9v} \) can be broken down as \( \sqrt{9} \cdot \sqrt{v} = 3 \sqrt{v} \).
• Similarly, \( \sqrt{25v} \) simplifies to \( \sqrt{25} \cdot \sqrt{v} = 5 \sqrt{v} \).

Essentially, you simplify the numerical part first and then rewrite the expression with the variable under the square root.

Combining Like Terms

Combining like terms is essential to simplifying expressions. Terms are considered 'like' if they have the same variables raised to the same powers. When working with square roots:

• First, ensure all terms have the same square root component.
• In this case, both simplified terms have \( \sqrt{v} \): \( 9x \sqrt{v} \) and \( 20 \sqrt{v} \).

However, to combine these further, their coefficients and variable parts must match exactly. Here, the expressions cannot be combined into one term but must remain as they are due to different coefficients (\(9x\) and \(20\)). Combining like terms often involves adding or subtracting their coefficients, but only if they are like.

Coefficients

Coefficients are the numerical parts of terms in expressions. They play a significant role when simplifying expressions with square roots.

• In the expression \( 3x \sqrt{9v} \), '3x' is the coefficient.
• Once the square root is simplified, it becomes \( 3x \cdot 3 \sqrt{v} = 9x \sqrt{v} \).
• Similarly, simplify \( 4 \sqrt{25v} \) to \( 4 \cdot 5 \sqrt{v} = 20 \sqrt{v} \).

Like in the simplified terms, when you multiply a coefficient by the simplified square root, you adjust the numerical part accordingly. Coefficients are crucial, as they provide the scale for combining or factoring the expressions.