Question

3\(x^{3} \cdot 2 x^{2} y \cdot 4 x^{2} y\) is equivalent to: F. 9\(x^{7} y^{2}\) G. 9\(x^{12} y^{2}\) H. 24\(x^{7} y^{2}\) J. 24\(x^{12} y\) K. 24\(x^{12} y^{2}\)

Short answer

Question: Simplify the expression 3\(x^{3} \cdot 2 x^{2} y \cdot 4 x^{2} y\) and choose the equivalent expression from the options below. A) 12\(x^{7}y^{2}\) B) 12\(x^{7}y\) C) 24\(x^{7}y\) D) 24\(x^{6}y^{2}\) E) 24\(x^{6}y\) F) 24\(x^{5}y^{2}\) G) 24\(x^{5}y\) H) 24\(x^{7}y^{2}\)

Step-by-step solution

Multiply the coefficients

Multiply the coefficients together: 3 * 2 * 4 = 24

Combine and simplify the terms with the same base

Apply the product rule to simplify the terms with the same base: \(x^{3} \cdot x^{2} \cdot x^{2} = x^{(3 + 2 + 2)} = x^{7}\) and \(y\cdot y = y^2\)

Combine simplified expressions

Combine the results from Step 1 and Step 2: 24\(x^{7}y^{2}\) By breaking down the exercise and simplifying the expression, we can see that the correct answer is choice H, which is equal to 24\(x^{7}y^{2}\).

Key concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. These expressions are the foundation of algebra and can represent real-world scenarios in mathematical terms.

Take the example of the expression from the exercise, 3\(x^{3} \cdot 2 x^{2} y \cdot 4 x^{2} y\). It consists of coefficients (3, 2, and 4), variables (\(x\) and \(y\)), and exponents (3 and 2), joined by multiplication signs. To solve such expressions, we perform operations such as addition, subtraction, multiplication, and division among terms.

Understanding how to manipulate these expressions is crucial, as it forms the basis for more complex problem-solving such as equation solving, inequalities, and functions analysis. Simplifying algebraic expressions often requires combining like terms, which are terms that have the same variables raised to the same power, as per the exercise we're examining.

Exponent Rules

Exponent rules, also known as the laws of exponents, dictate how to handle mathematical expressions involving powers. Essential exponent rules include:

• The Product Rule: \(a^m \cdot a^n = a^{m+n}\)
• The Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\) when \(neq0\)
• The Power Rule: \((a^m)^n = a^{m\cdot n}\)
• Power of a Product Rule: \((ab)^m = a^m \cdot b^m\)
• Power of a Quotient Rule: \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\) when \(beq0\)

These rules help simplify expressions and solve equations that feature exponents. They are fundamental to effectively working through algebraic problems. For instance, in the given exercise, we used the product rule to combine powers of the same base, \(x\).

Product Rule for Exponents

The product rule for exponents is a critical tool for simplifying expressions when multiplying variables with the same base. It states that when you multiply like bases, you add the exponents. The mathematical representation is: \(a^m \cdot a^n = a^{m+n}\).

For example, in the original exercise, we multiply \(x^{3} \cdot x^{2} \cdot x^{2}\). Since all terms have the base \(x\), we add their exponents: 3 + 2 + 2 to get \(x^{7}\). Similarly, for \(y\cdot y\), we have the same base \(y\), so we add the exponents, 1 + 1, to achieve \(y^2\).

This simple yet powerful rule allows us to combine multiple exponential terms efficiently and is a foundation piece in algebraic manipulation. It is especially useful in higher-level mathematics, where expressions become more complex.