Problem 39 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ 3 x^{2} \cdot 2 x^{5} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(3x^{2} \cdot 2x^{5}\) using the product rule of exponents. Answer: The simplified expression is \(6x^7\).

Step 1: Write the given expression

Write down the given expression: $$ 3x^{2} \cdot 2x^{5} $$

Step 2: Apply the product rule of exponents

Since the bases are the same (x), we can apply the product rule of exponents, which states that when the bases are the same, we simply add the exponents. So we have: $$ (3 \cdot 2)(x^{(2 + 5)}) $$

Step 3: Simplify the numbers and exponents

Now, simplify the numbers and exponents: $$ (6)(x^7) $$

Step 4: Write the final simplified expression

The final simplified expression is: $$ 6x^7 $$ So, the simplified expression of the given problem using the product rule of exponents is \(6x^7\).

Key Concepts

Simplifying Expressions

Mastering the art of simplifying expressions is fundamental in algebra and serves as the cornerstone for solving complex equations. Simplification reduces expressions into their most manageable form, making them clearer and easier to work with.

Take the example of multiplying two algebraic terms with the same base: bringing together the coefficients and then employing the product rule for the exponents simplifies the work. In the equation \(3x^{2} \cdot 2x^{5}\), we're combining the constants, yielding \(3 \cdot 2\), and adding the exponents with the same base, resulting in \(x^{2 + 5}\). This use of simplification transforms the expression into a single term, \(6x^7\), a much neater and more understandable result.

While simplifying, always watch out for opportunities to combine like terms, use exponent rules, and reduce fractions if applicable. This process not only makes the math easier but also prepares you for more advanced concepts in algebra.

Exponent Rules

The Product Rule

The product rule is quintessential when dealing with exponents. It states that when you multiply two powers with the same base, you can add the exponents while keeping the base the same: \[x^{a} \cdot x^{b} = x^{a+b}\]. In the example \(3x^{2} \cdot 2x^{5}\), applying the product rule seamlessly merges the exponents of 'x' leading to \(x^{2+5}\) or \(x^7\).

Additional Rules

Besides the product rule, there are other exponent rules to remember: the quotient rule dictates that dividing powers with the same base lets you subtract the exponents \[x^{a} \/ x^{b} = x^{a-b}\], and the power rule \(x^{a})^{b} = x^{ab}\) for taking a power to a power. Being versed in these rules is instrumental in tackling algebra with confidence and efficiency.

Algebraic Expressions

Algebraic expressions are the phrases of the algebraic language, comprised of numbers, variables, and arithmetic operations. They are fundamental in conveying mathematical ideas and solving problems. The expression \(3x^{2} \cdot 2x^{5}\) is a product of two such expressions, combining numerical coefficients and variables with exponents.

An adeptness in handling algebraic expressions implies not just carrying out arithmetic operations, but also recognizing and applying algebraic structures and rules, like the aforementioned exponent rules. The goal is to manipulate these expressions to reveal simpler or more insightful forms, as in transforming \(3x^{2} \cdot 2x^{5}\) to \(6x^7\), which is a much tidier and practical representation in algebraic terms.