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Fraction multiplication involves multiplying the numerators (top numbers) and the denominators (bottom numbers) of two fractions. For example, in the exercise, we need to multiply \(\frac{2}{5}\) by both \(\frac{1}{3} p\) and \(\frac{7}{10} w\). Here’s a step-by-step breakdown of how we do it:

- Multiply the numerators: \(\frac{2}{5} \times \frac{1}{3}\) becomes \(\frac{2 \times 1}{5 \times 3} = \frac{2}{15}\)
- Similarly, for \(\frac{2}{5} \times \frac{7}{10}\), multiply numerators and denominators: \(\frac{2 \times 7}{5 \times 10} = \frac{14}{50} = \frac{7}{25}\).

Practice makes perfect with fraction multiplication. Remember to simplify the result if possible. In our example, \(\frac{14}{50}\) simplifies to \(\frac{7}{25}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

Combining like terms means adding or subtracting coefficients of terms that have the same variable raised to the same power. For instance, in our exercise, the terms \(\frac{2}{15}p\) and \(\frac{7}{25}w\) are distinct because they involve different variables. To combine like terms, you:

- Identify terms with the same variable and power
- Add or subtract their coefficients

In expressions where no like terms exist, as in our example, you simply leave them as they are. Always pay attention to the variables and their exponents to correctly combine like terms. This skill is crucial in simplifying algebraic expressions and solving equations.

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They represent quantities and their relationships. In our given exercise, the expression is \(\frac{2}{5}\big(\frac{1}{3}p + \frac{7}{10}w\big)\). Here’s a brief walkthrough of key concepts:

- **Variables:** Symbols like \(p\) and \(w\) that stand for unknown values
- **Constants:** Numbers on their own, like 2, 5, 1, 3, 7, and 10
- **Coefficients:** The numerical part of the terms with variables. In our example, \(\frac{1}{3}\) and \(\frac{7}{10}\) are coefficients of \(p\) and \(w\)

To simplify algebraic expressions, distribute and combine like terms as shown in the exercise solution. Becoming comfortable with these will make it easier to solve more complex equations and handle various algebraic operations.