Problem 50
Question
Simplify to make the calculation as easy as possible. \(16 \cdot(-31)+16 \cdot 32\)
Step-by-Step Solution
Verified Answer
The simplified expression is 16.
1Step 1: Identify Common Factors
Both terms in the expression \(16 \cdot (-31) + 16 \cdot 32\) share a common factor of 16. To simplify the expression, identify the common factor which can be factored out.
2Step 2: Factor Out the Common Factor
Factor out the common factor of 16. This gives us: \(16((-31) + 32)\).
3Step 3: Simplify Inside the Parentheses
Now, we need to simplify the expression inside the parentheses. Calculate \((-31) + 32\) which results in 1.
4Step 4: Perform the Simplification
Substitute the simplified result from step 3 back into the expression: \(16 \cdot 1\). Since multiplying by 1 does not change the value, the expression simplifies to 16.
Key Concepts
SimplificationFactoringArithmetic Operations
Simplification
Simplification is a method used in math to make solving problems easier. It involves reducing expressions to their simplest form by combining like terms, removing parentheses, and applying basic arithmetic operations.
To simplify an expression, look for patterns or shortcuts that reduce the number of operations you need to perform. This often includes identifying common factors or terms that can be combined. For the expression \(16 \cdot(-31)+16\cdot 32\), simplifying means spotting that both terms are multiplied by 16.
Recognizing such common factors allows us to restructure the problem into an easier form. This makes calculations faster and reduces errors.
To simplify an expression, look for patterns or shortcuts that reduce the number of operations you need to perform. This often includes identifying common factors or terms that can be combined. For the expression \(16 \cdot(-31)+16\cdot 32\), simplifying means spotting that both terms are multiplied by 16.
Recognizing such common factors allows us to restructure the problem into an easier form. This makes calculations faster and reduces errors.
Factoring
Factoring involves breaking down an expression into a product of its factors. It is a crucial skill in prealgebra and higher levels of math.
In the exercise \(16\cdot(-31) + 16\cdot 32\), both terms are multiplied by 16. Factoring out the 16 means we rewrite the expression as a single product: \(16((-31) + 32)\).
Factoring simplifies expressions and helps in solving equations. You can often find the greatest common factor and use it to transform complicated expressions into manageable steps. This is especially helpful in reducing long computations and spotting opportunities for further reduction.
In the exercise \(16\cdot(-31) + 16\cdot 32\), both terms are multiplied by 16. Factoring out the 16 means we rewrite the expression as a single product: \(16((-31) + 32)\).
Factoring simplifies expressions and helps in solving equations. You can often find the greatest common factor and use it to transform complicated expressions into manageable steps. This is especially helpful in reducing long computations and spotting opportunities for further reduction.
Arithmetic Operations
Arithmetic operations are basic calculations like addition, subtraction, multiplication, and division. These operations are the building blocks of math.
Once the expression \(16((-31) + 32)\) has been factored, the next step is to simplify inside the parentheses using arithmetic operations. This involves calculating \((-31) + 32\), which equals 1.
Finally, the expression becomes \(16 \cdot 1\). Since multiplying by 1 leaves any number unchanged, the result is simply 16. This illustrates how arithmetic operations can be used step by step to reach the simplest result in a problem.
Once the expression \(16((-31) + 32)\) has been factored, the next step is to simplify inside the parentheses using arithmetic operations. This involves calculating \((-31) + 32\), which equals 1.
Finally, the expression becomes \(16 \cdot 1\). Since multiplying by 1 leaves any number unchanged, the result is simply 16. This illustrates how arithmetic operations can be used step by step to reach the simplest result in a problem.
Other exercises in this chapter
Problem 50
Use the Distributive Property to write each expression as an equivalent algebraic expression. $$(e+f)(-5)$$
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Solve each equation. Check your solution $$15 s=75$$
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Simplify each expression. $$3-4(m+1)$$
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The solution of which equation is NOT graphed on the number line below? \(\mathbf{F} 12=-6 x\) \(\mathbf{H}-14=7 x\) G \(8 x=-16 \quad\) J \(-18 x=-36\)
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