Problem 114 - Elementary Algebra

Question

Simplify the following problems. $$ a^{3} b^{7} \cdot \frac{a^{9} b^{6}}{a^{5} b^{10}} $$

Step-by-Step Solution

Answer

Based on the given step-by-step solution, explain how the expression \(a^{3}b^{7}\cdot \frac{a^{9}b^{6}}{a^{5}b^{10}} \) is simplified to \(a^{7}b^{3}\). The expression is simplified by using the Product of Powers Rule and the Quotient of Powers Rule for exponents. First, the Product of Powers Rule is applied to combine the terms in the numerator, resulting in the simplified numerator of \(a^{12}b^{13}\). Then, the Quotient of Powers Rule is used to simplify the expression by dividing the exponents of like bases. This results in an expression of \(a^{7}b^{3}\), which is the final simplified answer.

Step 1: Rewrite the expression

We first rewrite the given expression as follows: $$ a^{3}b^{7}\cdot \frac{a^{9}b^{6}}{a^{5}b^{10}} $$

Step 2: Combine the terms in the numerator

We use the product of powers rule, which states \(a^{m}a^{n}=a^{m+n}\), to combine the terms in the numerator: $$ \left(a^{3}b^{7}\right) \left(a^{9}b^{6}\right) $$ This simplifies to: $$ a^{3+9}b^{7+6} $$ which is: $$ a^{12}b^{13} $$

Step 3: Rewrite the expression with the product of powers

We now rewrite the expression with the product of the powers in the numerator and the original denominator: $$ \frac{a^{12}b^{13}}{a^{5}b^{10}} $$

Step 4: Apply the Quotient of Powers Rule

To simplify the expression, we apply the quotient of powers rule, which states that \(\frac{a^{m}}{a^{n}}=a^{m-n}\). Doing so, we get: $$ a^{12-5}b^{13-10} $$ Which simplifies to: $$ a^{7}b^{3} $$ The final simplified expression is: $$ a^{7}b^{3} $$

Key Concepts

Exponent Rules

Understanding exponent rules is essential when dealing with algebraic expressions that include powers. These rules are a set of guidelines that describe how to handle mathematical operations involving exponents. One of the fundamental rules is that when you multiply numbers with the same base, you add their exponents. Another rule involves dividing powers with the same base, where you subtract the exponent in the denominator from the exponent in the numerator.

These basic exponent rules also apply to variables. For instance, if you have the expression \(a^{m}\) multiplied by \(a^{n}\), our result would be \(a^{m+n}\), where \(a\) is the base and \(m\) and \(n\) are the exponents. Similarly, the rule for dividing is applied as \(\frac{a^{m}}{a^{n}} = a^{m-n}\). These simple yet powerful rules streamline the process of simplifying algebraic expressions.

Product of Powers Rule

Applying the Product of Powers Rule

When simplifying algebraic expressions that involve multiplication of the same base, the product of powers rule comes to the rescue. This rule states: if you multiply two powers with the same base, keep the base unchanged and add the exponents together.

Here's a step-by-step application of this rule:

• Consider the expression \(a^{m} \times a^{n}\).
• The base \(a\) remains the same.
• Simply add the exponents: \(m + n\).
• The simplified expression will be \(a^{m+n}\).

When we encounter expressions with multiple variables, such as \(a^{3}b^{7} \times a^{9}b^{6}\), we apply this rule separately to each variable resulting in \(a^{3+9}b^{7+6}\), leading to further simplification.

Quotient of Powers Rule

Understanding the Quotient of Powers Rule

Dividing expressions with exponents can be simplified using the quotient of powers rule. This rule states that when you divide powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator. Like the product of powers rule, it applies to algebraic expressions that contain variables.

If we have \(\frac{a^{m}}{a^{n}}\), the steps to simplify using this rule would be:

• Keep the base \(a\) constant.
• Subtract the exponent \(n\) in the denominator from the exponent \(m\) in the numerator.
• Write the result as \(a^{m-n}\).

For a more complex expression such as \(\frac{a^{12}b^{13}}{a^{5}b^{10}}\), apply the rule to each variable to get \(a^{12-5}b^{13-10}\), which simplifies to \(a^{7}b^{3}\).

Algebraic Simplification

Practical Tips for Algebraic Simplification

Algebraic simplification is the process of reducing expressions to their simplest form using various algebraic rules and properties. The goal is to make the expression as concise as possible without changing its value. Simplification might involve combining like terms, applying exponent rules, factoring, expanding expressions, or utilizing properties of operations.

To simplify an algebraic expression, systematically apply the rules of arithmetic and algebra. For instance, after applying product and quotient of powers rules, ensure there are no like terms that can be combined further. Always double-check your work for accuracy; simplification errors can lead to incorrect solutions in algebra. It's important to not only understand the rules but to also practice applying them in multiple scenarios to become proficient in algebraic simplification.