Problem 34 - Elementary and Intermediate Algebra

Question

Use the product rule for exponents to simplify each expression. Write the results using exponents. $$ \left(u^{3} v^{5}\right)\left(u^{4} v^{5}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( u^{7} v^{10} \).

Step 1: Apply the Product Rule for Exponents

The product rule states that when multiplying two powers with the same base, you add the exponents. For the expression \( \left(u^{3} v^{5}\right)\left(u^{4} v^{5}\right) \), start by identifying the like bases: \( u \) and \( v \).

Step 2: Simplify the Exponents for 'u'

Apply the product rule to the base \( u \). Since both terms have \( u \), combine the exponents by adding them: \( u^{3} \times u^{4} = u^{3+4} = u^{7} \).

Step 3: Simplify the Exponents for 'v'

Apply the product rule to the base \( v \). Since both terms have \( v \), combine the exponents by adding them: \( v^{5} \times v^{5} = v^{5+5} = v^{10} \).

Step 4: Write the Final Expression

Combine the simplified terms with their new exponents: \( u^{7} v^{10} \). This is the simplified form of the original expression using exponents.

Key Concepts

Product Rule for Exponents

The product rule for exponents is a fundamental principle in algebra that simplifies expressions involving exponents. It states that when you multiply two powers that have the same base, you simply add their exponents. This rule can be expressed mathematically as:

• If you have \( a^m \times a^n \), the result is \( a^{m+n} \).

It's essential to identify the base in each term of the expression. In this exercise, the bases are \( u \) and \( v \). Both are raised to different powers, so we apply the rule separately to each base. By following this method, complex expressions can be simplified into more manageable terms. This rule helps in organizing and reducing expressions, making them easier to work with in equations.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra to make equations more manageable. The process involves reducing an expression to its simplest form. It can include combining like terms, using arithmetic, and applying rules for exponents. Let's look at how we apply it.

• Identify like terms: Look for variables raised to exponents that can be combined. In our exercise, \( u^3 \) and \( u^4 \) are like terms because they share the base \( u \), and \( v^5 \) terms can be combined as well.
• Apply rules: Use the product rule for exponents to add the exponents of like terms.

The final step of combining the expressions involves writing the equation in a simpler form. This makes future calculations easier and clearer, avoiding errors in more complex equations.

Exponent Rules

Understanding exponent rules is essential for working with powers effectively in algebra.There are several rules that one must become familiar with:

• Product Rule: As discussed, to multiply terms with the same base, add the exponents.
• Power Rule: To raise a power to another power, multiply the exponents: \((a^m)^n = a^{m\cdot n}\).
• Quotient Rule: When dividing terms with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

In our expression \( \left(u^{3} v^{5}\right)\left(u^{4} v^{5}\right) \), recognizing these rules allows us to seamlessly combine terms and solve more complicated problems. By practicing these rules, you can become more adept at handling problems involving powers, ensuring accuracy in simplifying and solving equations.