Which Expression Is Equivalent To $144^{\frac{3}{2}}$?A. 216 B. 1728 C. $\sqrt[3]{12}$ D. $ 72 3 \sqrt[3]{72} 3 72 ​ [/tex]

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Understanding the Problem

When dealing with exponents and roots, it's essential to understand the properties and rules that govern them. In this problem, we're given the expression $144^{\frac{3}{2}}$ and asked to find an equivalent expression from the given options. To solve this, we need to apply the rules of exponents and roots.

Applying the Rules of Exponents and Roots

The expression $144^{\frac{3}{2}}$ can be rewritten using the rule of exponents that states $a^{m/n} = \sqrt[n]{a^m}$. Applying this rule, we get:

14432=14432144^{\frac{3}{2}} = \sqrt[2]{144^3}

Simplifying the Expression

Now, let's simplify the expression inside the square root. We know that $144 = 12^2$, so we can rewrite the expression as:

(122)32\sqrt[2]{(12^2)^3}

Using the rule of exponents that states $(am)n = a^{mn}$, we can simplify the expression further:

122⋅32\sqrt[2]{12^{2 \cdot 3}}

1262\sqrt[2]{12^6}

Evaluating the Square Root

Now, let's evaluate the square root of $12^6$. We know that $12^6 = (122)3 = 144^3$, so we can rewrite the expression as:

14432\sqrt[2]{144^3}

Finding the Equivalent Expression

From the given options, we need to find an expression that is equivalent to $144^{\frac{3}{2}}$. Let's analyze each option:

A. 216: This option is not equivalent to $144^{\frac{3}{2}}$.

B. 1728: This option is equivalent to $144^{\frac{3}{2}}$.

C. $\sqrt[3]12}$ This option is not equivalent to $144^{\frac{3{2}}$.

D. $\sqrt[3]72}$ This option is not equivalent to $144^{\frac{3{2}}$.

Conclusion

Based on our analysis, we can conclude that the expression equivalent to $144^{\frac{3}{2}}$ is option B. 1728.

Understanding the Concept of Exponents and Roots

Exponents and roots are fundamental concepts in mathematics that are used to represent repeated multiplication and division. In this problem, we applied the rules of exponents and roots to simplify the expression $144^{\frac{3}{2}}$ and find an equivalent expression.

Real-World Applications of Exponents and Roots

Exponents and roots have numerous real-world applications in fields such as science, engineering, and finance. For example, in physics, exponents and roots are used to describe the behavior of particles and waves. In engineering, exponents and roots are used to design and optimize systems. In finance, exponents and roots are used to calculate interest rates and investment returns.

Tips for Solving Exponent and Root Problems

When solving exponent and root problems, it's essential to understand the properties and rules that govern them. Here are some tips to help you solve these types of problems:

  • Understand the concept of exponents and roots and how they are used to represent repeated multiplication and division.
  • Apply the rules of exponents and roots to simplify expressions and find equivalent expressions.
  • Use the properties of exponents and roots to evaluate expressions and solve problems.
  • Practice solving exponent and root problems to build your skills and confidence.

Common Mistakes to Avoid

When solving exponent and root problems, it's essential to avoid common mistakes. Here are some common mistakes to avoid:

  • Not understanding the concept of exponents and roots and how they are used to represent repeated multiplication and division.
  • Not applying the rules of exponents and roots to simplify expressions and find equivalent expressions.
  • Not using the properties of exponents and roots to evaluate expressions and solve problems.
  • Not practicing solving exponent and root problems to build your skills and confidence.

Conclusion

In conclusion, the expression equivalent to $144^{\frac{3}{2}}$ is option B. 1728. Understanding the concept of exponents and roots and how they are used to represent repeated multiplication and division is essential to solving these types of problems. By applying the rules of exponents and roots and using the properties of exponents and roots, you can simplify expressions and find equivalent expressions. With practice and patience, you can build your skills and confidence in solving exponent and root problems.

Q: What is the difference between an exponent and a root?

A: An exponent is a number that represents repeated multiplication, while a root is a number that represents repeated division. For example, $2^3 = 8$ and $\sqrt[3]{8} = 2$.

Q: How do I simplify an expression with an exponent?

A: To simplify an expression with an exponent, you can use the rules of exponents, such as $a^{m/n} = \sqrt[n]{a^m}$ and $(am)n = a^{mn}$.

Q: How do I evaluate an expression with a root?

A: To evaluate an expression with a root, you can use the definition of a root, which is $\sqrt[n]{a} = b$ if and only if $b^n = a$.

Q: What is the order of operations for exponents and roots?

A: The order of operations for exponents and roots is the same as for regular arithmetic operations: parentheses, exponents, roots, multiplication and division, and addition and subtraction.

Q: Can I simplify an expression with both an exponent and a root?

A: Yes, you can simplify an expression with both an exponent and a root by using the rules of exponents and roots. For example, $\sqrt[3]{2^4} = \sqrt[3]{16} = 2.5198421$.

Q: How do I solve an equation with an exponent or a root?

A: To solve an equation with an exponent or a root, you can use the rules of exponents and roots to isolate the variable. For example, $2^x = 8$ can be solved by using the rule $a^{m/n} = \sqrt[n]{a^m}$ to get $x = \log_2 8$.

Q: What are some common mistakes to avoid when working with exponents and roots?

A: Some common mistakes to avoid when working with exponents and roots include:

  • Not understanding the concept of exponents and roots and how they are used to represent repeated multiplication and division.
  • Not applying the rules of exponents and roots to simplify expressions and find equivalent expressions.
  • Not using the properties of exponents and roots to evaluate expressions and solve problems.
  • Not practicing solving exponent and root problems to build your skills and confidence.

Q: How can I practice solving exponent and root problems?

A: You can practice solving exponent and root problems by working through exercises and problems in a textbook or online resource. You can also try solving real-world problems that involve exponents and roots, such as calculating interest rates or investment returns.

Q: What are some real-world applications of exponents and roots?

A: Exponents and roots have numerous real-world applications in fields such as science, engineering, and finance. For example, in physics, exponents and roots are used to describe the behavior of particles and waves. In engineering, exponents and roots are used to design and optimize systems. In finance, exponents and roots are used to calculate interest rates and investment returns.

Q: How can I use exponents and roots in my daily life?

A: You can use exponents and roots in your daily life by applying the rules of exponents and roots to solve problems that involve repeated multiplication and division. For example, you can use exponents and roots to calculate interest rates on loans or investments, or to determine the growth rate of a population.

Q: What are some common types of exponent and root problems?

A: Some common types of exponent and root problems include:

  • Simplifying expressions with exponents and roots
  • Evaluating expressions with exponents and roots
  • Solving equations with exponents and roots
  • Calculating interest rates and investment returns using exponents and roots

Q: How can I improve my skills in solving exponent and root problems?

A: You can improve your skills in solving exponent and root problems by practicing regularly and working through exercises and problems in a textbook or online resource. You can also try solving real-world problems that involve exponents and roots, and seek help from a teacher or tutor if you need it.

Q: What are some resources for learning more about exponents and roots?

A: Some resources for learning more about exponents and roots include:

  • Textbooks and online resources that cover exponents and roots
  • Online tutorials and videos that explain exponents and roots
  • Practice problems and exercises that help you build your skills in solving exponent and root problems
  • Real-world applications of exponents and roots in fields such as science, engineering, and finance.