Problem 42 - 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. $$ 2 a^{3} b^{2} \cdot 3 a b $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(2a^{3}b^{2} \cdot 3ab\). Answer: \(6a^{4}b^{3}\)

Step 1: Apply the product rule to the terms with the same base

Given the expression $$2a^{3}b^{2} \cdot 3ab$$, we have $$a^{3} \cdot a$$ and $$b^{2} \cdot b$$. Using the product rule, we get: \(a^{3} \cdot a = a^{3+1} = a^{4}\) \(b^{2} \cdot b = b^{2+1} = b^{3}\) Now we can rewrite the expression as $$2a^{4}b^{3} \cdot 3$$.

Step 2: Combine the numbers in the expression

Now that we have simplified the exponents, we have $$2a^{4}b^{3} \cdot 3$$. We can combine the numbers \(2\) and \(3\) by multiplying them together: \(2 \cdot 3 = 6\)

Step 3: Write the final expression

We now have all the simplified components and can rewrite the final expression: \(6a^{4}b^{3}\) So $$2a^{3}b^{2} \cdot 3ab$$ simplifies to $$6a^{4}b^{3}$$.

Key Concepts

Quotient Rule of Exponents

The quotient rule of exponents is essential when dividing terms that have the same base. It makes it easy to simplify expressions by handling the exponents directly. When you have a fraction with the same base in both the numerator and denominator, the rule is to subtract the exponent of the denominator from that of the numerator: \[ a^{m} / a^{n} = a^{m-n} \]This handy rule allows you to shrink expressions by reducing the powers of like bases. The expression, for example, \( x^{5}/x^{2} \) would simplify to \( x^{3} \). Here, you subtract the exponents, resulting in \( 5 - 2 = 3\). This can often simplify fractions significantly, leading to more straightforward calculations.

Simplifying Expressions

Simplifying expressions is the process of rewriting them in the most condensed form possible, while retaining the same value. The aim is to make complex algebraic expressions easier to manage. In our problem, simplifying involved multiplying coefficients and applying the product rule to combine terms with like bases. Some strategies include:

• Combining like terms
• Applying rules of exponents, such as the product rule
• Factoring expressions when possible

Breaking down expressions allows for better understanding and sets the stage for solving more complex equations. Simplification aids in reducing errors in calculations as expressions become easier to handle. Each step in the simplification process should adhere to algebraic principles to ensure accuracy.

Exponents

Exponents indicate how many times a number, known as the base, is used in a multiplication.Using the example \(a^{3}\), this means you multiply the base \(a\) by itself thrice: \(a \times a \times a\). Exponents follow specific rules that govern their manipulation:

• Product Rule: When multiplying like bases, add their exponents: \( a^{m} \cdot a^{n} = a^{m+n} \)
• Quotient Rule: When dividing like bases, subtract the exponent of the denominator: \( a^{m} / a^{n} = a^{m-n} \)
• Power Rule: When raising a power to a power, multiply the exponents: \( (a^{m})^{n} = a^{mn} \)

These rules are vital for simplifying algebraic expressions and solving equations. Proper mastery of these rules allows for quick and accurate simplification of expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations, such as addition or multiplication. They do not contain an equals sign, which differentiates them from algebraic equations. Understanding how to manipulate these expressions is key to solving algebra problems.Elements of algebraic expressions include:

• Terms: Parts of the expression separated by '+' or '−' signs, like \(2a^{3}\) or \(3b\)
• Coefficients: The numerical part of a term, such as 2 in \(2a^{3}\)
• Variables: Symbols representing unknown values, like \(a\) and \(b\)
• Exponents: Numbers that show how many times the base is multiplied by itself, such as 3 in \(a^{3}\)

When tackling algebraic expressions, identifying and understanding these components allows students to simplify or solve them more effectively. Manipulating expressions often involves applying methods and rules of arithmetic and algebra to reach a more straightforward or desirable form.