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Chapter 1: Problem 13

Are the expressions equivalent? $$ x^{2}+4 x^{2}, 5 x^{2}, \text { and } 4 x^{4} $$

Short Answer

Expert verified
Answer: The first and second expressions are equivalent, but the third expression is not equivalent to the others.

Step by step solution

01

Simplify the first expression

In the first expression, both terms have the same base, \(x\), and exponent, \(2\). So we can combine the coefficients and add them: $$ x^2 + 4x^2 = 1x^2 + 4x^2 = 5x^2 $$
02

Compare the first expression with the second expression

We have simplified the first expression to be \(5x^2\). Now, we have: 1. \(5x^2\) 2. \(5x^2\) As we can see, these two expressions are equal to each other.
03

Compare the second expression with the third expression

Now we have only to compare the expressions \(5x^2\) and \(4x^4\). They have different exponents; specifically, the first one has the exponent \(2\) while the second one has the exponent \(4\). Due to this difference, they cannot be equivalent.
04

Conclusion

The first and second expressions are equivalent, both being simplified to \(5x^2\). However, the third expression, \(4x^4\), is not equivalent to the others.

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Key Concepts

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

Equivalence of Expressions
To determine if two algebraic expressions are equivalent, we need to simplify each expression to its simplest form, then compare the results.
Equivalence in algebra means that two expressions will yield the same value for all values of the variables involved. Let's break this down:
  • If two expressions simplify down to the same form, they are equivalent.
  • Checking equivalence helps in solving equations and verifying mathematical identities.
In our example, we had the expressions: - \(x^2 + 4x^2\) - \(5x^2\) - \(4x^4\)After simplifying, the first expression becomes \(5x^2\), which matches exactly with the second. However, \(4x^4\) is not equivalent because it represents a different form due to its exponent, as we'll see in the following sections.
Simplifying Expressions
Simplifying expressions is the process of combining like terms to make an expression easier to understand and use. Let's explore how this works:
  • Start with identifying like terms, which are terms that have the same variable raised to the same power.
  • In \(x^2 + 4x^2\), both terms are like terms.
  • You can add their coefficients to simplify: \(1x^2 + 4x^2 = 5x^2\).
Simplification is crucial because it allows expressions to be easily compared and solved, reduces the complexity of equations, and eases the process of finding solutions. In our exercise, by simplifying \(x^2 + 4x^2\) to \(5x^2\), we made it easier to establish equivalence with the other expressions.
Exponents
Exponents are used in algebra to denote repeated multiplication of a number by itself. Here's what you need to know:
  • The exponent indicates how many times the base is multiplied by itself.
  • A term like \(x^2\) means \(x\) multiplied by itself, which is \(x \times x\).
  • In our problem, we had terms with exponents like \(x^2\) and \(x^4\).
Understanding exponents is essential when comparing expressions. In our exercise, the terms \(5x^2\) and \(4x^4\) appear different due to their exponents. Even with identical coefficients, different exponents change the value and meaning of the expression.
This difference made it impossible for \(5x^2\) and \(4x^4\) to be equivalent. Recognizing these differences allows for proper assessment of equivalence between algebraic expressions.

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Most popular questions from this chapter

Evaluate and simplify the expressions in Problems 44-53. $$ h(2 x-b-3) \text { given } h(x)=2 x-b-2 \text { . } $$

Given that \(x=4\) is a solution to $$ 2 j x+z=3 $$ evaluate the expression \(16 j+2 z\).

Evaluate and simplify the expressions in Problems 44-53. $$ f(5) \text { given } f(t)=\frac{t+1}{t-2} . $$

It costs a contractor \(\$ p\) to employ a plumber, \(\$ e\) to employ an electrician, and \(\$ c\) to employ a carpenter. (a) Write an expression for the total cost to employ 4 plumbers, 3 electricians, and 9 carpenters. (b) Write an expression for the fraction of the total cost in part (a) that is due to plumbers. (c) Suppose the contractor hires \(P\) plumbers, \(E\) electricians, and \(C\) carpenters. Write expressions for the total cost for hiring these workers and the fraction of this cost that is due to plumbers.

Find the average rate of change of \(f(x)=x^{3}-x^{2}\) on the intervals indicated in Exercises 28-30. Between 2 and \(4 .\)
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