Problem 57 - Elementary Algebra

Question

Simplify \(\left(4 x^{3} y^{8}\right)\left(3 x^{2} y\right)\)

Step-by-Step Solution

Answer

Question: Simplify the expression \((4x^3y^8)(3x^2y)\). Answer: The simplified expression is \(12x^5y^9\).

Step 1: Identify the coefficients and variables with their exponents

The given expression is \((4x^3y^8)(3x^2y)\). Here, the coefficients are \(4\) and \(3\), and the variables are \(x\) and \(y\). The exponents of \(x\) are \(3\) and \(2\), and the exponent of \(y\) is \(8\) and \(1\) for the two terms, respectively.

Step 2: Multiply the coefficients

Multiply the coefficients \(4\) and \(3\) together: \(4\times 3 = 12\). This will be the coefficient for our simplified expression.

Step 3: Adding exponents of like terms

For the \(x\) terms, we have \(x^3\) and \(x^2\). Add the exponents: \(3 + 2 = 5\). The simplified \(x\) term is \(x^5\). For the \(y\) terms, we have \(y^8\) and \(y\). Since the second \(y\) term does not have an explicit exponent, it is assumed to be \(1\). Add the exponents: \(8 + 1 = 9\). The simplified \(y\) term is \(y^9\).

Step 4: Write the simplified expression

Combine the simplified coefficient and variables with their exponents: \(12x^5y^9\). This is the simplified expression for \((4x^3y^8)(3x^2y)\).

Key Concepts

Coefficients

In an algebraic expression, a coefficient is the numerical factor that multiplies a variable or a product of variables. Coefficients are crucial because they determine the magnitude or size of the terms in expressions. For instance, in the expression \(4x^3y^8\), the number \(4\) is the coefficient, while in \(3x^2y\), the coefficient is \(3\).

• The coefficients in the context of an algebraic problem determine how many times the variables and their associated powers are counted.
• To simplify expressions that involve coefficients, you simply multiply them together if they are part of a multiplication operation. For example, \(4\) and \(3\) are multiplied to give \(12\), the coefficient in the final expression.

Keep in mind that coefficients can also be negative or fractional, impacting the direction or division of the term they modify.

Exponents

Exponents are a fundamental part of algebraic expressions, representing the number of times a variable is multiplied by itself. An exponent is written as a superscript next to the base variable.In the expression \(x^3\), \(3\) is the exponent, indicating that \(x\) is used as a factor three times: \(x \times x \times x\).

• Exponents follow specific rules when expressions are simplified, particularly the Product of Powers property. This property states that when multiplying like bases, you can add the exponents: \(x^m \times x^n = x^{m+n}\).
• For example, in the exercise \((4x^3y^8)(3x^2y)\), you combine exponents as follows: add exponents \(3\) and \(2\) for \(x\), resulting in \(x^5\), and \(8\) and \(1\) for \(y\), resulting in \(y^9\).

Understanding how exponents work allows for efficient simplification of complex expressions.

Simplifying Expressions

Simplifying expressions is the process of reducing them to simpler but equivalent expressions. This process helps in solving equations more easily and efficiently.In simplifying the expression \((4x^3y^8)(3x^2y)\), you follow a series of steps:

• Multiply Coefficients: Start by multiplying the numerical coefficients \(4\) and \(3\) to get \(12\).
• Add Exponents: For like variables, add the exponents using the Product of Powers rule: \(x^3 \times x^2 = x^{3+2} = x^5\) and \(y^8 \times y = y^{8+1} = y^9\).
• Combine Elements: Put it all together to form the simplified expression, which is \(12x^5y^9\).

Simplification basically involves applying arithmetic operations and algebraic rules to make expressions more concise and manageable for further calculations or applications.