Problem 21 - Algebra for College Students

Question

Simplify each numerical expression. Be sure to take advantage of the properties whenever they can be used to make the computations easier. $$17(97)+17(3)$$

Step-by-Step Solution

Answer

The simplified expression is 1700.

Step 1: Identify the Common Factor

In the expression \(17(97) + 17(3)\), both terms share a common factor, which is 17.

Step 2: Use the Distributive Property

Rewrite the expression by factoring out the common factor: \(17(97) + 17(3) = 17(97 + 3)\). This step uses the distributive property: \(a(b+c) = ab + ac\).

Step 3: Simplify Inside the Parentheses

Calculate the sum inside the parentheses: \(97 + 3 = 100\).

Step 4: Multiply the Factors

Multiply the factored number by the sum from step 3: \(17 \, \times \, 100\).

Step 5: Compute the Final Result

Compute the multiplication: \(17 \, \times \, 100 = 1700\).

Key Concepts

Distributive Property

When simplifying algebraic or numerical expressions, the distributive property is a helpful tool. It allows us to multiply a single term by two or more terms inside a parenthesis. Essentially, it states that for any numbers or expressions, if you have something like \( a(b + c) \), you can distribute \( a \) across the terms inside the parenthesis. This leads to \( ab + ac \).

Using the distributive property can make calculations more straightforward, especially when dealing with large numbers or complex expressions. For example, in the expression \( 17(97) + 17(3) \), instead of doing two separate multiplications involving 17, you can factor out the 17 by using the distributive property to get \( 17(97 + 3) \). This consolidates the operation and makes it easier to compute the final result.

Factoring

Factoring is a crucial concept in algebra, and it refers to rewriting an expression as a product of its factors. When expressions have common factors, you can simplify them by factoring these out. This is a particularly valuable technique for reducing larger or more complex expressions to simpler forms.

In the exercise, the expression \( 17(97) + 17(3) \) can be factored by taking out the common factor of 17, resulting in \( 17(97 + 3) \). This means that both parts of the expression originally involved a multiplication by 17, and by factoring, we make the expression easier to handle, moving from a detailed two-step calculation to a more compact one-step calculation.

Numerical Expressions

Numerical expressions involve numbers and operations like addition, subtraction, multiplication, and division. Simplifying these expressions involves applying mathematical properties to reduce them to simpler forms.

In this exercise, we dealt with the numerical expression \( 17(97) + 17(3) \). Step-by-step simplification using the distributive property and factoring led us from a more complicated form into \( 17(100) \), which then easily evaluates to 1700. Understanding how to handle numerical expressions effectively allows for simpler and faster calculations, which is particularly useful in solving problems efficiently.