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Chapter 0: Problem 50

Simplify each expression. $$ -4+4(4 k-3)-6(2 k+8)+7 $$

Short Answer

Expert verified
4k - 57

Step by step solution

01

Distribute the Constants

First, distribute the constants in the expression to simplify it. Apply the distributive property: Distribute 4: Distribute -6: -6(2k + 8) = -12k - 48 The expression is now: -4 + 16k - 12 - 12k - 48 + 7
02

Combine Like Terms

Combine the like terms in the expression: Combine the constants Combine the k-terms 16k - 12k = 4k Therefore, the expression simplifies to: 4k - 57

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Key Concepts

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

Distributive Property
The distributive property is a key concept in algebra which helps to simplify expressions. It works by distributing the coefficient outside the parenthesis to each term within the parenthesis. This is done by multiplying the outside term with each inside term.
In the given problem, we distribute two constants: 4 and -6.

Here's how the property is used:
  • Distribute 4 through the expression \(4k - 3\):
    4 \times 4k \rightarrow 16k
    4 \times -3 \rightarrow -12

  • Distribute -6 through the expression \(2k + 8\):
    -6 \times 2k \rightarrow -12k
    -6 \times 8 \rightarrow -48
After distribution, the expression looks like this:
onumber -4 + 16k - 12 - 12k - 48 + 7
The distributive property helps in breaking down complex expressions, making it easier to proceed with further simplifications.

It's essential in algebra to distribute terms correctly to avoid errors in the final simplified form.
Combining Like Terms
Combining like terms is a method for simplifying algebraic expressions, making them easier to understand and solve. Like terms are terms that have the same variable part raised to the same power. Only their coefficients differ.
In the given expression:
onumber -4 + 16k - 12 - 12k - 48 + 7
we group terms that have similar characteristics:
  • k-terms: 16k and -12k
  • Constants: -4, -12, -48, and 7

Combining k-terms first:
16k - 12k = 4k
Then, combining constants:
onumber -4 - 12 - 48 + 7 = -57
After combining like terms, the simplified expression is:
4k - 57
By combining like terms, we reduce the complexity of the expression, leading to a simpler and more manageable form to work with.
Constants
Constants in algebra refer to fixed numerical values that do not change. Unlike variables, which can represent multiple values, constants retain the same value.
In the provided expression: onumber -4 + 16k - 12 - 12k - 48 + 7
we identify the constants:
  • onumber -4
  • onumber -12
  • onumber -48
  • onumber 7
The constants are combined to form a single constant value by performing standard arithmetic operations:
onumber -4 - 12 - 48 + 7 = -57
Understanding constants is crucial because they provide a fixed point of reference in algebraic expressions. Correctly handling constants, particularly in combination with variables, is vital for accurately simplifying and solving expressions.

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Most popular questions from this chapter

Evaluate each expression for \(w=4, x=-\frac{3}{4}, y=\frac{1}{2},\) and \(z=1.25 .\) $$ w y-8 x $$

Evaluate each expression for \(a=-3, b=64,\) and \(c=6 .\) $$ \frac{a^{3}+2 c}{-\sqrt{b}} $$

Evaluate each expression for \(w=4, x=-\frac{3}{4}, y=\frac{1}{2},\) and \(z=1.25 .\) $$ w-6 x+5 y-3 z $$

Simplify each inequality if needed. Then determine whether the statement is true or false. $$ -6<7+3 $$

Simplify each inequality if needed. Then determine whether the statement is true or false. $$ 2 \cdot 5 \geq 4+6 $$
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