Simplify The Expression $\left(14 F G^2 H^2\right)\left(3 F^4 G^2 H^2\right$\]. Write The Variables In Alphabetical Order.

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying the given expression $\left(14 f g^2 h^2\right)\left(3 f^4 g^2 h^2\right)$ and write the variables in alphabetical order. We will break down the process into manageable steps, making it easy to follow and understand.

Understanding the Expression

The given expression is a product of two terms, each containing variables and constants. To simplify the expression, we need to apply the rules of exponents and combine like terms.

$\left(14 f g^2 h^2\right)\left(3 f^4 g^2 h^2\right)$

Step 1: Apply the Product Rule for Exponents

When multiplying two terms with the same base, we add the exponents. In this case, we have two terms with the same base, $f$, $g$, and $h$. We will apply the product rule for exponents to simplify the expression.

$\left(14 f g^2 h^2\right)\left(3 f^4 g^2 h^2\right) = 14 \cdot 3 \cdot f \cdot f^4 \cdot g^2 \cdot g^2 \cdot h^2 \cdot h^2$

Step 2: Simplify the Constants

We will multiply the constants, 14 and 3, to get the new constant.

$14 \cdot 3 = 42$

Step 3: Apply the Product Rule for Exponents (continued)

Now, we will apply the product rule for exponents to the variables. We have two terms with the same base, $f$, $g$, and $h$. We will add the exponents.

$f \cdot f^4 = f^{1+4} = f^5$ $g^2 \cdot g^2 = g^{2+2} = g^4$ $h^2 \cdot h^2 = h^{2+2} = h^4$

Step 4: Combine the Simplified Terms

Now, we will combine the simplified terms to get the final expression.

$42 f^5 g^4 h^4$

Writing the Variables in Alphabetical Order

To write the variables in alphabetical order, we will rearrange the terms.

$f^5 g^4 h^4$

Conclusion

In this article, we simplified the given expression $\left(14 f g^2 h^2\right)\left(3 f^4 g^2 h^2\right)$ and wrote the variables in alphabetical order. We applied the product rule for exponents and combined like terms to simplify the expression. We also rearranged the variables to write them in alphabetical order. This process demonstrates the importance of algebraic manipulation in mathematics and provides a step-by-step guide for simplifying complex expressions.

Frequently Asked Questions

• Q: What is the product rule for exponents? A: The product rule for exponents states that when multiplying two terms with the same base, we add the exponents.
• Q: How do I simplify an expression with variables and constants? A: To simplify an expression with variables and constants, you need to apply the product rule for exponents and combine like terms.
• Q: How do I write variables in alphabetical order? A: To write variables in alphabetical order, you need to rearrange the terms in the expression.

Final Answer

The final answer is $\boxed{42 f^5 g^4 h^4}$.

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying the given expression $\left(14 f g^2 h^2\right)\left(3 f^4 g^2 h^2\right)$ and write the variables in alphabetical order. We will break down the process into manageable steps, making it easy to follow and understand.

Understanding the Expression

The given expression is a product of two terms, each containing variables and constants. To simplify the expression, we need to apply the rules of exponents and combine like terms.

$\left(14 f g^2 h^2\right)\left(3 f^4 g^2 h^2\right)$

Step 1: Apply the Product Rule for Exponents

When multiplying two terms with the same base, we add the exponents. In this case, we have two terms with the same base, $f$, $g$, and $h$. We will apply the product rule for exponents to simplify the expression.

$\left(14 f g^2 h^2\right)\left(3 f^4 g^2 h^2\right) = 14 \cdot 3 \cdot f \cdot f^4 \cdot g^2 \cdot g^2 \cdot h^2 \cdot h^2$

Step 2: Simplify the Constants

We will multiply the constants, 14 and 3, to get the new constant.

$14 \cdot 3 = 42$

Step 3: Apply the Product Rule for Exponents (continued)

Now, we will apply the product rule for exponents to the variables. We have two terms with the same base, $f$, $g$, and $h$. We will add the exponents.

$f \cdot f^4 = f^{1+4} = f^5$ $g^2 \cdot g^2 = g^{2+2} = g^4$ $h^2 \cdot h^2 = h^{2+2} = h^4$

Step 4: Combine the Simplified Terms

Now, we will combine the simplified terms to get the final expression.

$42 f^5 g^4 h^4$

Writing the Variables in Alphabetical Order

To write the variables in alphabetical order, we will rearrange the terms.

$f^5 g^4 h^4$

Conclusion

In this article, we simplified the given expression $\left(14 f g^2 h^2\right)\left(3 f^4 g^2 h^2\right)$ and wrote the variables in alphabetical order. We applied the product rule for exponents and combined like terms to simplify the expression. We also rearranged the variables to write them in alphabetical order. This process demonstrates the importance of algebraic manipulation in mathematics and provides a step-by-step guide for simplifying complex expressions.

Frequently Asked Questions

Q&A Section

Q: What is the product rule for exponents?

A: The product rule for exponents states that when multiplying two terms with the same base, we add the exponents.

Q: How do I simplify an expression with variables and constants?

A: To simplify an expression with variables and constants, you need to apply the product rule for exponents and combine like terms.

Q: How do I write variables in alphabetical order?

A: To write variables in alphabetical order, you need to rearrange the terms in the expression.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that does not change.

Q: How do I apply the product rule for exponents?

A: To apply the product rule for exponents, you need to multiply the exponents of the same base.

Q: What is the order of operations in algebra?

A: The order of operations in algebra is Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction (PEMDAS).

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, you need to apply the product rule for exponents and combine like terms.

Q: What is the difference between a coefficient and a variable?

A: A coefficient is a number that is multiplied by a variable, while a variable is a letter or symbol that represents a value that can change.

Final Answer

The final answer is $\boxed{42 f^5 g^4 h^4}$.