Problem 19 - Precalculus: Functions and Graphs

Question

Simplify. $$\left(3 u^{7} v^{3}\right)\left(4 u^{4} v^{-5}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(12 u^{11} v^{-2}\).

Step 1: Multiply the Coefficients

First, multiply the numerical coefficients of the terms: \(3 \times 4 = 12\). Thus, we have:\[12 u^{7} v^{3} u^{4} v^{-5}\]

Step 2: Apply the Product of Powers Property to 'u'

According to the product of powers property, when multiplying like bases, add the exponents: \(u^{7} \times u^{4} = u^{7+4} = u^{11}\). The expression now becomes:\[12 u^{11} v^{3} v^{-5}\]

Step 3: Apply the Product of Powers Property to 'v'

Similarly, apply the product of powers property for the base 'v': \(v^{3} \times v^{-5} = v^{3+(-5)} = v^{-2}\). The expression now simplifies to:\[12 u^{11} v^{-2}\]

Step 4: Final Simplified Expression

The simplified expression is obtained by combining the results from previous steps:\[12 u^{11} v^{-2}\]

Key Concepts

Product of Powers Property

When working with algebraic expressions involving exponents, the Product of Powers Property is a vital concept to understand. This property states that when you multiply two powers with the same base, you can add their exponents. This is instrumental in simplifying expressions efficiently.
For example, consider the expression \(a^m \times a^n\). According to the product of powers property, this expression simplifies to \(a^{m+n}\).

• If you have \(u^7 \times u^4\), you add the exponents: \(7 + 4\), which results in \(u^{11}\).
• Similarly, the base \(v\) can be simplified with \(v^3 \times v^{-5} = v^{3 + (-5)} = v^{-2}\).

Understanding this property makes it easier to work with complex expressions and is frequently used in algebra.

Simplifying Expressions

Simplifying expressions is a method to make an expression more manageable or easier to understand by reducing it to its simplest form. This often involves combining like terms, applying arithmetic operations, and utilizing properties of exponents.In the expression \((3u^7v^3)(4u^4v^{-5})\), the initial step is to deal with the coefficients separately. Multiply the numbers: \(3 \times 4 = 12\), which gives the simplified coefficient.
Next, you handle each variable separately using the product of powers property. By doing this, the expression becomes more concise: \(12u^{11}v^{-2}\). Remember, simplifying helps solve equations more quickly and prevents errors in longer calculations.

Exponents

Exponents are a shorthand way of expressing repeated multiplication of a number by itself. For instance, \(a^3\) means \(a\) is multiplied by itself three times: \(a \times a \times a\). This notation helps simplify processes and calculations.Key things to remember about exponents:

• When multiplying with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
• If you have a power raised to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).
• Any number raised to the power of zero is 1: \(a^0 = 1\) for any \(a eq 0\).

Exponents make it easier to write and work with large numbers and expressions.

Negative Exponents

Negative exponents can be a tricky concept, but they are actually quite simple once you understand the rule: a negative exponent indicates that the base is on the opposite side of a fraction. This means \(a^{-n} = \frac{1}{a^n}\).

• In the expression \(v^{-2}\), it implies \(\frac{1}{v^2}\). Applying this helps simplify expressions where variables or numbers appear to the power of a negative exponent.
• Understanding negative exponents also aids in converting them to positive exponents when expressions need to be rewritten or solved further.

Remember, simplifying negative exponents is crucial in ensuring clarity and accuracy in algebraic expressions.