Problem 9 - Elementary Algebra

Question

Perform each multiplication in one step. $$ x^{4} \cdot 4 y^{2} \cdot 2 x^{2} \cdot 7 y^{6} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$4x^4y^2 * 2x^2y^6 * 7$$ in one step. Answer: After simplifying the expression in one step, the result is $$56 x^{6} y^{8}$$.

Step 1: Multiply the constants together

First, we will multiply the constants 4, 2 and 7 together: 4 * 2 * 7 = 56

Step 2: Use the properties of exponents

Now, we will use the properties of exponents to simplify the expression. Recall that when multiplying terms with the same base, we add the exponents: \(x^{4} \cdot x^{2} = x^{(4+2)} = x^{6}\) Likewise for y: \(y^{2} \cdot y^{6} = y^{(2+6)} = y^{8}\)

Step 3: Combine the simplified expressions

Finally, we will combine the simplified expressions from Steps 1 and 2: 56 * \(x^{6}\) * \(y^{8}\) So, after performing the multiplication in one step, the simplified expression is: $$ 56 x^{6} y^{8} $$

Key Concepts

Properties of Exponents

When dealing with expressions that involve exponents, it is crucial to understand the properties that govern their operations. Exponents are used to signify repeated multiplication of a number by itself. Here are some fundamental properties:

• Product of Powers: When multiplying like bases, you add the exponents, i.e., \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power: When raising an exponent to another power, multiply the exponents, i.e., \((a^m)^n = a^{m \cdot n}\).
• Zero Exponent: Any base raised to the power of zero equals one, i.e., \(a^0 = 1\) \((a eq 0)\).
• Negative Exponents: An expression with a negative exponent denotes reciprocal, i.e., \(a^{-m} = \frac{1}{a^m}\).

In the given exercise, we see the application of the product of powers property. By recognizing common bases \(x\) and \(y\), their respective exponents are added together: \(x^4 \cdot x^2 = x^{6}\) and \(y^2 \cdot y^6 = y^{8}\). These properties are foundational in simplifying expressions efficiently.

Simplifying Expressions

Simplifying expressions is the process of reducing an algebraic expression to its most compact and comprehensible form. The main objective is to perform operations in a way that retains the expression's integrity while making it easier to understand or solve.

• Identify Like Terms: Like terms are terms that have identical variables raised to the same power. Only like terms can be combined.
• Apply Mathematical Operations: Use arithmetic operations such as addition, subtraction, multiplication, and division systematically. In the context of exponent use, ensure that you adhere strictly to the properties of exponents.
• Reorder and Group Terms: Arrange terms in a logical order that often helps to quickly identify like terms or factors.

In the given exercise, the expression \(x^4 \cdot 4y^2 \cdot 2x^2 \cdot 7y^6\) is simplified. By multiplying the constants and applying the properties of exponents, we're left with a tidier expression: \(56x^6y^8\). This step reduces complexity and prepares the expression for further operations, if needed.

Algebraic Multiplication

Algebraic multiplication involves applying multiplication to algebraic terms, which may include variables, constants, or coefficients.
Multiplying algebraic expressions primarily involves three steps:

• Multiplying Coefficients: Coefficients (the numerical part of the terms) are straightforward to multiply. In our example, 4, 2, and 7 are multiplied to give 56.
• Applying Exponential Rules: When variables are involved, it is important to handle them with care, using known exponent rules to ensure accuracy, as seen in \(x^4 \cdot x^2 = x^6\) and \(y^2 \cdot y^6 = y^8\).
• Combining After Multiplication: After coefficients and variables are multiplied separately, they are combined into one unified product, i.e., \(56x^6y^8\).

It is customary to write expressions in a standard form where terms are ordered by decreasing powers of a variable, and multiplied products are condensed accurately.This approach aids students in not only solving particular exercises like the one given, but also equips them with general problem-solving skills they can apply to more complex algebraic operations.