Simplify The Expression: 16 A 4 B 12 4 = □ \sqrt[4]{16 A^4 B^{12}} = \square 4 16 A 4 B 12 ​ = □

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. One of the most common ways to simplify expressions is by using the properties of exponents and roots. In this article, we will focus on simplifying the expression $\sqrt[4]{16 a^4 b^{12}}$ using these properties.

Understanding the Expression

The given expression is a fourth root of a product of two terms: $16 a^4 b^{12}$. To simplify this expression, we need to understand the properties of fourth roots and exponents. The fourth root of a number is a value that, when raised to the power of 4, gives the original number. In other words, $\sqrt[4]{x} = y$ means that $y^4 = x$.

Breaking Down the Expression

To simplify the expression, we can break it down into two parts: the coefficient and the variables. The coefficient is the numerical part of the expression, which is $16$. The variables are the algebraic parts, which are $a^4$ and $b^{12}$.

Simplifying the Coefficient

The coefficient $16$ can be simplified by expressing it as a power of $2$, since $16 = 2^4$. Therefore, we can rewrite the expression as $\sqrt[4]{(2^4) a^4 b^{12}}$.

Simplifying the Variables

The variables $a^4$ and $b^{12}$ can be simplified by using the property of exponents that states $a^m \cdot a^n = a^{m+n}$. Since both $a^4$ and $b^{12}$ have the same base, we can combine them by adding their exponents. Therefore, we can rewrite the expression as $\sqrt[4]{(2^4) (a^4)^1 (b^{12})}$.

Applying the Property of Roots

Now that we have broken down the expression into its coefficient and variables, we can apply the property of roots that states $\sqrt[n]{a^n} = a$. Since the exponent of $a$ is $4$, we can simplify the expression by taking the fourth root of $a^4$, which is equal to $a$. Similarly, since the exponent of $b$ is $12$, we can simplify the expression by taking the fourth root of $b^{12}$, which is equal to $b^3$.

Simplifying the Expression

Now that we have applied the property of roots, we can simplify the expression by combining the simplified variables and the coefficient. Therefore, we can rewrite the expression as $\sqrt[4]{(2^4) (a^4)^1 (b^{12})} = 2 a b^3$.

Conclusion

In this article, we simplified the expression $\sqrt[4]{16 a^4 b^{12}}$ using the properties of exponents and roots. We broke down the expression into its coefficient and variables, simplified the coefficient and variables, and applied the property of roots to simplify the expression. The final simplified expression is $2 a b^3$.

Frequently Asked Questions

• Q: What is the property of roots that we used to simplify the expression? A: The property of roots that we used is $\sqrt[n]{a^n} = a$.
• Q: How do we simplify the coefficient $16$? A: We simplify the coefficient $16$ by expressing it as a power of $2$, since $16 = 2^4$.
• Q: How do we simplify the variables $a^4$ and $b^{12}$? A: We simplify the variables $a^4$ and $b^{12}$ by using the property of exponents that states $a^m \cdot a^n = a^{m+n}$.

Final Answer

The final answer is: $\boxed{2 a b^3}$

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Q&A: Simplifying the Expression

Q: What is the property of roots that we used to simplify the expression?

A: The property of roots that we used is $\sqrt[n]{a^n} = a$. This property states that the nth root of a number raised to the power of n is equal to the number itself.

Q: How do we simplify the coefficient $16$?

A: We simplify the coefficient $16$ by expressing it as a power of $2$, since $16 = 2^4$. This allows us to rewrite the expression as $\sqrt[4]{(2^4) a^4 b^{12}}$.

Q: How do we simplify the variables $a^4$ and $b^{12}$?

A: We simplify the variables $a^4$ and $b^{12}$ by using the property of exponents that states $a^m \cdot a^n = a^{m+n}$. Since both $a^4$ and $b^{12}$ have the same base, we can combine them by adding their exponents. Therefore, we can rewrite the expression as $\sqrt[4]{(2^4) (a^4)^1 (b^{12})}$.

Q: What is the final simplified expression?

A: The final simplified expression is $2 a b^3$. This is obtained by applying the property of roots and simplifying the variables.

Q: Can we simplify the expression further?

A: No, the expression $2 a b^3$ is already in its simplest form. We have applied all the necessary properties of exponents and roots to simplify the expression.

Q: How do we know when to apply the property of roots?

A: We apply the property of roots when we have a number raised to a power that is equal to the index of the root. In this case, we had $a^4$ and $b^{12}$, which are both raised to powers that are equal to the index of the fourth root.

Q: Can we use the property of roots to simplify other expressions?

A: Yes, the property of roots can be used to simplify other expressions as well. For example, if we have the expression $\sqrt[3]{27 x^3 y^9}$, we can simplify it by applying the property of roots and simplifying the variables.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not applying the property of roots when it is applicable
• Not simplifying the variables correctly
• Not combining like terms
• Not checking the final expression for errors

Q: How do we check the final expression for errors?

A: We can check the final expression for errors by plugging in some values for the variables and checking if the expression holds true. We can also use algebraic manipulations to check if the expression is equivalent to the original expression.

Q: Can we use technology to simplify expressions?

A: Yes, technology can be used to simplify expressions. For example, we can use calculators or computer algebra systems to simplify expressions and check for errors.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including:

• Physics: Simplifying expressions is used to solve problems in physics, such as calculating the trajectory of a projectile.
• Engineering: Simplifying expressions is used to design and optimize systems, such as electronic circuits.
• Economics: Simplifying expressions is used to model and analyze economic systems, such as supply and demand curves.

Q: Can we use simplifying expressions to solve problems in other fields?

A: Yes, simplifying expressions can be used to solve problems in other fields, such as:

• Computer science: Simplifying expressions is used to optimize algorithms and data structures.
• Biology: Simplifying expressions is used to model and analyze biological systems, such as population growth.
• Chemistry: Simplifying expressions is used to model and analyze chemical reactions.

Q: What are some tips for simplifying expressions?

A: Some tips for simplifying expressions include:

• Breaking down the expression into smaller parts
• Identifying and applying the property of roots
• Simplifying the variables correctly
• Combining like terms
• Checking the final expression for errors

Q: Can we use simplifying expressions to solve problems in mathematics?

A: Yes, simplifying expressions is a fundamental skill in mathematics, and it is used to solve problems in many areas of mathematics, including algebra, geometry, and calculus.