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Understanding exponents is crucial in algebra. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \(3^2\), the base is 3 and the exponent is 2. This means: \(3^2 = 3 \times 3 = 9\). When dealing with variables, you treat them similarly. For instance, \((x)^3\) means \(x \times x \times x = x^3\). When you see an expression like \((3x)^2\), both the coefficient (3) and the variable (x) are affected by the exponent. So it becomes \(3^2 \times x^2 = 9x^2\). Exponents make it easier to represent repeated multiplication succinctly.

Multiplication of variables follows a straightforward rule: you multiply the coefficients and then apply the laws of exponents to the variables. Let's break down the process using our example from the exercise: \((3x)^2 \times (5x)\). First, handle the exponent as: \((3x)^2 = 3^2 \times x^2 = 9x^2\). Next, multiply this result with \(5x\). Multiply the coefficients: \(9 \times 5 = 45\), and then the variables with their exponents: \(x^2 \times x = x^{2+1} = x^3\). So, the product is \(45x^3\). Always keep the coefficients and variable parts separate initially and combine them at the end for clarity.

Like terms are terms in an expression that have identical variable parts. Only their coefficients can be different. For example, \(3x^2\) and \(7x^2\) are like terms, but \(5x\) and \(5y\) are not. To combine like terms, you can simply add or subtract their coefficients. In multiplication, like terms combine by adding their exponents. For example, in the expression \(8y^3 \times 6y\), both terms have the variable \(y\). When you multiply these, you add the exponents of \(y\): \(y^3 \times y^1 = y^{3+1} = y^4\). The coefficients multiply as usual: \(8 \times 6 = 48\). Therefore, the expression simplifies to \(48y^4\). Recognizing and combining like terms efficiently is key to solving and simplifying algebraic expressions.