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Chapter 1: Problem 70

Simplify each expression. $$ -3(2 t+4)+8(2 t-4) $$

Short Answer

Expert verified
Simplified expression: \( 10t - 44 \)

Step by step solution

01

- Distribute the constants

Apply the distributive property to both terms in the expression. For the first term, distribute \( -3 \) to \( 2t \) and \( 4 \): \( -3(2t) \) and \( -3(4) \). For the second term, distribute \( 8 \) to \( 2t \) and \( -4 \): \( 8(2t) \) and \( 8(-4) \).
02

- Simplify the distributed terms

Calculate each distribution: \( -3(2t) = -6t \), \( -3(4) = -12 \), \( 8(2t) = 16t \), and \( 8(-4) = -32 \). This results in a new expression: \( -6t - 12 + 16t - 32 \).
03

- Combine like terms

Combine the terms with \( t \): \( -6t + 16t \) and the constant terms: \( -12 - 32 \). This results in \( (-6t + 16t) + (-12 - 32) \).
04

- Simplify further

Simplify the combined terms: \( -6t + 16t = 10t \) and \( -12 - 32 = -44 \). Therefore, the simplified expression is \( 10t - 44 \).

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Key Concepts

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

Distributive Property
Let's understand the distributive property, one of the core principles in algebra. It allows you to multiply a single term by each term within a parenthesis. This property is represented as: \[a(b + c) = ab + ac\]
This can make complex expressions simpler.
For instance, applying this to the expression \(-3(2t + 4)\):
  • Multiply \(-3\) by \(2t\): \(-3 \times 2t = -6t\)
  • Multiply \(-3\) by 4: \(-3 \times 4 = -12\)
The same process applies to the second part of the expression: \(8(2t - 4)\).
It becomes:
  • 8 times \(2t\): \(8 \times 2t = 16t\)
  • 8 times \(-4\): \(8 \times -4 = -32\)
You have successfully distributed the constants over the terms in the parentheses. The intermediate expression now becomes: \(-6t -12 + 16t - 32\).
Combining Like Terms
After applying the distributive property, the next step is to combine like terms. Like terms have the same variable raised to the same power.
A good practice is to group these terms together, making it easier to simplify.
In our example, we combine like terms from \(-6t -12 + 16t - 32\):
  • Group the terms with \(t\): \(-6t\) and \(16t\)
  • Group the constants: \(-12\) and \(-32\)
Combining these, we first add \(-6t\) and \(16t\) to get: \[-6t + 16t = 10t\]
Next, we add the constants: \[-12 - 32 = -44\]
Now, our expression is simplified further to: \[10t - 44\].
Algebraic Simplification
Simplifying algebraic expressions is about making them as concise as possible.
This involves applying properties like distributive property and combining like terms.
Simplification helps in better understanding and solving equations.
From our previous steps:
  • We used the distributive property to expand: \(-6t - 12 + 16t - 32\)
  • We combined like terms to get: \(10t - 44\)
This final expression \(10t - 44\) is much simpler and easier to handle.
Understanding these simplification techniques is essential. It helps greatly in solving more complex equations and problems in algebra.
Remember, practice makes perfect. The more you engage with these properties, the more intuitive it will become!

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Most popular questions from this chapter

Write a numerical expression for each phrase and simplify. The quotient of \(-12\) and the sum of \(-5\) and \(-1\)

The operation of division is used in divisibility tests. A divisibility test allows us to determine whether a given number is divisible (without remainder) by another number. An integer is divisible by 12 if it is divisible by both 3 and \(4,\) and not otherwise. Show that (a) \(4,249,474\) is not divisible by 12 (b) \(4,253,520\) is divisible by 12 and

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) $$ \frac{x y+8 a}{x-y} $$

Translate each phrase into a mathematical expression. Use \(x\) as the variable. Combine like terms when possible. Five times a number, added to the sum of the number and three

Write a numerical expression for each phrase and simplify. Three-fourths of the sum of \(-8\) and 12
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