Problem 128 - Intermediate Algebra

Question

Explain how the distributive property can be used to evaluate the expression \(52.713(21)+52.713(79)\) mentally.

Step-by-Step Solution

Answer

The expression simplifies to \(5271.3\).

Step 1: Identify Common Factor

Look at the expression \(52.713(21) + 52.713(79)\). Notice that the number \(52.713\) is a common factor in both terms. Both terms can be combined by using the distributive property.

Step 2: Factor Out the Common Term

Use the distributive property to factor out the common factor from both terms. The expression becomes \(52.713(21 + 79)\). This means you are factoring \(52.713\) out of the expression.

Step 3: Simplify Inside the Parentheses

Now add the numbers inside the parentheses: \(21 + 79 = 100\). So the expression simplifies to \(52.713 \times 100\).

Step 4: Evaluate the Expression Mentally

Finally, multiply \(52.713\) by \(100\). When multiplying by 100, you simply move the decimal point two places to the right: \(52.713 \times 100 = 5271.3\).

Key Concepts

Factoring Common Term

To efficiently solve mathematical expressions like \(52.713(21) + 52.713(79)\), we can use the technique of factoring a common term. This technique is part of the distributive property, which in simple terms means re-distributing a factor to different terms within an expression.

• First, recognize the common factor in each term of the expression. In our example, the term \(52.713\) is present in both parts of the expression.
• By factoring out \(52.713\), you essentially rewrite the expression in terms of multiplication: \(52.713(21 + 79)\).

This step helps simplify what might initially look cumbersome and breaks the problem into more digestible parts. Instead of handling two separate operations, you are only left with one simplified expression.

Mental Math

Mental math involves calculating in your head without writing down interim steps. It’s a handy skill for simplifying expressions using techniques like the distributive property. For the expression \(52.713(21) + 52.713(79)\), mental math can streamline the process significantly.

• Once you factor out the common term, you are left with a much simpler problem: \(52.713(100)\), because \(21 + 79 = 100\).
• Multiplying by 100 is a very straightforward calculation as you simply move the decimal point two places to the right for decimal numbers.

This process reduces complexity and allows you to quickly reach the final solution, \(5271.3\), all in your head without needing paper or calculator. Mental math, therefore, leverages your understanding of numbers to make calculations faster and easier.

Simplifying Expressions

The ultimate goal of using concepts like the distributive property and factoring is to simplify expressions. Simplifying means rewriting an expression in a form that is easier to understand or calculate. For the expression \(52.713(21) + 52.713(79)\), the steps to simplify are:

• Identify and factor out the common term, here \(52.713\), to transform the expression into \(52.713(21 + 79)\).
• Perform any simple arithmetic like addition inside the parentheses first, simplifying to \(52.713(100)\).
• Finally, conduct the simplified multiplication of \(52.713 \times 100\), which results in \(5271.3\).

These steps not only make the expression quicker to evaluate but also aid in building a deeper understanding of underlying mathematical principles. By simplifying expressions, you often make them more accessible and solvable with minimal effort.