Simplify The Expression: 48 X 3 + 8 X 4 48x^3 + 8x^4 48 X 3 + 8 X 4

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Introduction

In algebra, simplifying expressions is a crucial step in solving equations and inequalities. It involves combining like terms and eliminating any unnecessary components. In this article, we will simplify the expression $48x^3 + 8x^4$ using various techniques.

Understanding the Expression

The given expression is a polynomial with two terms: $48x^3$ and $8x^4$. To simplify this expression, we need to understand the properties of exponents and like terms.

Properties of Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ means $x \times x \times x$. When we multiply two terms with the same base, we add their exponents. For instance, $x^3 \times x^2 = x^{3+2} = x^5$.

Like Terms

Like terms are terms that have the same variable and exponent. In the given expression, $48x^3$ and $8x^4$ are like terms because they both have the variable $x$ and an exponent. However, they have different coefficients (48 and 8) and exponents (3 and 4).

Simplifying the Expression

To simplify the expression, we need to combine the like terms. We can do this by adding the coefficients and keeping the same variable and exponent. In this case, we add 48 and 8, and keep the variable $x$ and exponent 4.

# Simplifying the expression
import sympy as sp
x = sp.symbols('x')
expr = 48x**3 + 8x**4
simplified_expr = sp.simplify(expr)
print(simplified_expr)

The simplified expression is $8x^4 + 48x^3$.

Factoring Out the Greatest Common Factor (GCF)

Another way to simplify the expression is to factor out the greatest common factor (GCF). The GCF of 48 and 8 is 8. We can factor out 8 from both terms:

# Factoring out the GCF
import sympy as sp
x = sp.symbols('x')
expr = 48x**3 + 8x4
gcf = 8
factored_expr = gcf * (6*x3 + x**4)
print(factored_expr)

The factored expression is $8(6x^3 + x^4)$.

Conclusion

In this article, we simplified the expression $48x^3 + 8x^4$ using various techniques. We combined like terms, added coefficients, and factored out the greatest common factor (GCF). The simplified expression is $8x^4 + 48x^3$, and the factored expression is $8(6x^3 + x^4)$. These techniques are essential in algebra and are used to simplify complex expressions.

Real-World Applications

Simplifying expressions has many real-world applications. In engineering, for example, simplifying expressions is used to design and optimize systems. In economics, simplifying expressions is used to model and analyze complex systems. In computer science, simplifying expressions is used to optimize algorithms and improve performance.

Common Mistakes

When simplifying expressions, there are several common mistakes to avoid. One mistake is to forget to combine like terms. Another mistake is to forget to factor out the greatest common factor (GCF). A third mistake is to simplify expressions incorrectly, leading to incorrect solutions.

Tips and Tricks

When simplifying expressions, here are some tips and tricks to keep in mind:

• Always combine like terms.
• Always factor out the greatest common factor (GCF).
• Use the distributive property to simplify expressions.
• Use the commutative property to simplify expressions.
• Use the associative property to simplify expressions.

Conclusion

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Introduction

In our previous article, we simplified the expression $48x^3 + 8x^4$ using various techniques. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

Q&A

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable and exponent. Unlike terms are terms that have different variables or exponents.

Q: How do I combine like terms?

A: To combine like terms, you need to add the coefficients and keep the same variable and exponent. For example, $2x^2 + 3x^2 = (2+3)x^2 = 5x^2$.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides two or more numbers. In the context of simplifying expressions, the GCF is used to factor out common factors from two or more terms.

Q: How do I factor out the GCF?

A: To factor out the GCF, you need to divide each term by the GCF. For example, if the GCF is 2, you would divide each term by 2: $4x^2 + 6x^2 = 2(2x^2) + 2(3x^2) = 2(2x^2 + 3x^2)$.

Q: What is the distributive property?

A: The distributive property is a property of multiplication that states that a single term can be multiplied by each term in a sum. For example, $a(b+c) = ab + ac$.

Q: How do I use the distributive property to simplify expressions?

A: To use the distributive property to simplify expressions, you need to multiply each term in the sum by the single term. For example, $2(x^2 + 3x) = 2x^2 + 6x$.

Q: What is the commutative property?

A: The commutative property is a property of addition and multiplication that states that the order of the terms does not change the result. For example, $a + b = b + a$ and $ab = ba$.

Q: How do I use the commutative property to simplify expressions?

A: To use the commutative property to simplify expressions, you need to rearrange the terms in the expression. For example, $x^2 + 3x = 3x + x^2$.

Q: What is the associative property?

A: The associative property is a property of addition and multiplication that states that the order in which you add or multiply terms does not change the result. For example, $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$.

Q: How do I use the associative property to simplify expressions?

A: To use the associative property to simplify expressions, you need to rearrange the terms in the expression. For example, $(x^2 + 3x) + 4x = x^2 + (3x + 4x)$.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to simplifying expressions. We covered topics such as like terms, unlike terms, combining like terms, the greatest common factor (GCF), factoring out the GCF, the distributive property, the commutative property, and the associative property. These properties and techniques are essential in algebra and are used to simplify complex expressions.

Real-World Applications

Simplifying expressions has many real-world applications. In engineering, for example, simplifying expressions is used to design and optimize systems. In economics, simplifying expressions is used to model and analyze complex systems. In computer science, simplifying expressions is used to optimize algorithms and improve performance.

Common Mistakes

When simplifying expressions, there are several common mistakes to avoid. One mistake is to forget to combine like terms. Another mistake is to forget to factor out the greatest common factor (GCF). A third mistake is to simplify expressions incorrectly, leading to incorrect solutions.

Tips and Tricks

When simplifying expressions, here are some tips and tricks to keep in mind:

• Always combine like terms.
• Always factor out the greatest common factor (GCF).
• Use the distributive property to simplify expressions.
• Use the commutative property to simplify expressions.
• Use the associative property to simplify expressions.

Conclusion

In conclusion, simplifying expressions is a crucial step in solving equations and inequalities. It involves combining like terms and eliminating any unnecessary components. In this article, we answered some frequently asked questions (FAQs) related to simplifying expressions. We covered topics such as like terms, unlike terms, combining like terms, the greatest common factor (GCF), factoring out the GCF, the distributive property, the commutative property, and the associative property. These properties and techniques are essential in algebra and are used to simplify complex expressions.