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Chapter 1: Problem 86

Use the distributive property to rewrite expression, then simplify. \(6(9+5)\)

Short Answer

Expert verified
The expression simplifies to 84 using the distributive property.

Step by step solution

01

Apply the Distributive Property

The distributive property states that for any numbers \(a, b,\) and \(c\), the expression \(a(b + c)\) can be rewritten as \(ab + ac\). Apply this to the expression \(6(9+5)\). Here, \(a = 6\), \(b = 9\), and \(c = 5\). This gives us:\[6(9+5) = 6 \cdot 9 + 6 \cdot 5\]
02

Multiply the Terms

Next, multiply each term by the number outside the parentheses. Calculate each product separately:\[6 \cdot 9 = 54\]\[6 \cdot 5 = 30\]
03

Simplify the Expression

Now that you have the products, add them together to simplify the expression:\[54 + 30 = 84\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rewriting Expressions
Rewriting expressions means changing the way an expression looks without altering its value. The distributive property is a useful tool for this task. It allows us to distribute a multiplier across terms inside a parenthesis. For instance, in the expression \(6(9+5)\), the number 6 is multiplied by both 9 and 5 separately. This technique ensures we accurately combine or reorganize expressions, making it easier to handle complex equations. By rewriting \(6(9+5)\) as \(6 \times 9 + 6 \times 5\), we make future calculations straightforward and manageable.
Simplifying Expressions
Simplifying expressions involves breaking them down into their simplest forms. This often means combining like terms and performing arithmetic operations to reduce the expression. After applying the distributive property, simplifying becomes straightforward. In our example with \(6(9+5)\), once rewritten as \(6 \times 9 + 6 \times 5\), simplifying involves calculating each multiplication separately.
  • Multiply 6 by 9 to get 54.
  • Multiply 6 by 5 to get 30.
  • Add these products: \(54 + 30\)
Finally, adding these results together gives the simplest form of the expression, which is 84.
Multiplication Basics
Understanding multiplication basics is crucial for handling expressions efficiently. At its core, multiplication is repeated addition. For instance, \(6 \times 9\) means adding 9 to itself 6 times. This foundational idea helps with applying the distributive property appropriately.

Consider \(6(9+5)\):
  • First, multiply 6 by 9.
  • Think of it as 9 repeated 6 times, giving 54.
  • Then, multiply 6 by 5 in a similar fashion, resulting in 30.
Once you grasp the basics of multiplication, applying it within expressions becomes intuitive, making more complex math problems easier to tackle.

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Most popular questions from this chapter

Find each of the following products. (Multiply.) In case use the shortcut method. $$\begin{array}{r}879 \\\\\times 600 \\\\\hline\end{array}$$

Write each of the following expressions in words. Use the word sum in each case. $$8+1$$

Write each of the following numbers in words. $$2,546,731$$

Round each of the numbers to the nearest hundred. $$5,044$$

Give the place value of each digit in the following numbers. $$768,253$$
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