Which Expression Is Equivalent To $144^{\frac{3}{2}}$?A. 216 B. 1,728 C. $\sqrt[3]{12}$ D. \$\sqrt[3]{72}$[/tex\]

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Introduction

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. When dealing with exponents, it's essential to understand how to simplify and manipulate them to find equivalent expressions. In this article, we'll explore the concept of simplifying exponents and apply it to the given expression $144^{\frac{3}{2}}$ to determine which of the provided options is equivalent.

Understanding Exponents

An exponent is a small number that is written above and to the right of a larger number, indicating how many times the larger number should be multiplied by itself. For example, in the expression $2^3$, the exponent 3 indicates that 2 should be multiplied by itself 3 times: $2^3 = 2 \times 2 \times 2 = 8$.

Simplifying Exponents

To simplify an exponent, we can use the following rules:

  • Product of Powers Rule: When multiplying two numbers with exponents, we add the exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.
  • Power of a Power Rule: When raising a number with an exponent to another power, we multiply the exponents. For example, $(23)4 = 2^{3 \times 4} = 2^{12}$.
  • Zero Exponent Rule: Any number raised to the power of 0 is equal to 1. For example, $2^0 = 1$.

Simplifying the Given Expression

Now, let's apply these rules to simplify the given expression $144^{\frac{3}{2}}$.

First, we can rewrite 144 as a product of its prime factors: $144 = 2^4 \times 3^2$.

Next, we can apply the Power of a Power Rule to simplify the exponent: $144^{\frac{3}{2}} = (2^4 \times 32){\frac{3}{2}} = 2^{4 \times \frac{3}{2}} \times 3^{2 \times \frac{3}{2}} = 2^6 \times 3^3$.

Now, we can simplify the expression further by evaluating the exponents: $2^6 \times 3^3 = 64 \times 27 = 1728$.

Comparing the Simplified Expression to the Options

Now that we have simplified the expression $144^{\frac{3}{2}}$ to $1728$, we can compare it to the provided options:

  • A. 216: This is not equivalent to the simplified expression.
  • B. 1728: This is equivalent to the simplified expression.
  • C. $\sqrt[3]{12}$: This is not equivalent to the simplified expression.
  • D. $\sqrt[3]{72}$: This is not equivalent to the simplified expression.

Conclusion

In conclusion, the expression $144^{\frac{3}{2}}$ is equivalent to $1728$. We simplified the expression using the rules of exponents and evaluated the exponents to arrive at the final answer. This demonstrates the importance of understanding and applying the rules of exponents to simplify complex expressions.

Final Answer

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

A: A base is the number being multiplied by itself, while an exponent is the number indicating how many times the base should be multiplied by itself.

Q: How do I simplify an exponent with a fraction?

A: To simplify an exponent with a fraction, you can use the Power of a Power Rule. For example, $2^{\frac{3}{2}} = \sqrt{2^3} = \sqrt{8}$.

Q: What is the rule for multiplying numbers with exponents?

A: When multiplying two numbers with exponents, you add the exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.

Q: What is the rule for raising a number with an exponent to another power?

A: When raising a number with an exponent to another power, you multiply the exponents. For example, $(23)4 = 2^{3 \times 4} = 2^{12}$.

Q: What is the rule for any number raised to the power of 0?

A: Any number raised to the power of 0 is equal to 1. For example, $2^0 = 1$.

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

A: To simplify an expression with multiple exponents, you can use the Product of Powers Rule and the Power of a Power Rule. For example, $2^3 \times 2^4 \times 2^5 = 2^{3+4+5} = 2^{12}$.

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

A: A radical is a symbol used to represent a root, while an exponent is a small number indicating how many times a base should be multiplied by itself.

Q: How do I simplify an expression with a radical and an exponent?

A: To simplify an expression with a radical and an exponent, you can use the Power of a Power Rule. For example, $\sqrt{2^3} = 2^{\frac{3}{2}}$.

Q: What is the final answer to the expression $144^{\frac{3}{2}}$?

A: The final answer to the expression $144^{\frac{3}{2}}$ is 1728.

Q: How do I apply the rules of exponents to simplify complex expressions?

A: To apply the rules of exponents to simplify complex expressions, you can use the Product of Powers Rule, the Power of a Power Rule, and the Zero Exponent Rule. For example, $2^3 \times 2^4 \times 2^5 = 2^{3+4+5} = 2^{12}$.

Conclusion

In conclusion, simplifying exponents is an essential skill in mathematics, and understanding the rules of exponents is crucial to simplifying complex expressions. By applying the Product of Powers Rule, the Power of a Power Rule, and the Zero Exponent Rule, you can simplify expressions with multiple exponents and arrive at the final answer.

Final Tips

  • Always start by simplifying the expression using the Product of Powers Rule.
  • Use the Power of a Power Rule to simplify expressions with multiple exponents.
  • Apply the Zero Exponent Rule to simplify expressions with a zero exponent.
  • Practice, practice, practice! The more you practice simplifying exponents, the more comfortable you'll become with the rules.