Solving 144^(3/2) A Step-by-Step Guide To Fractional Exponents

At its core, mathematics is a language, and exponents are a key part of its vocabulary. When we encounter expressions like $144^{\frac{3}{2}}$, it's crucial to understand the underlying principles that govern fractional exponents. This article will delve into the world of fractional exponents, dissecting the expression $144^{\frac{3}{2}}$ and guiding you step-by-step toward the correct solution. We will explore the fundamental rules of exponents and radicals, empowering you to confidently tackle similar problems in the future.

Understanding Fractional Exponents

Fractional exponents can appear daunting at first glance, but they are simply a shorthand way of expressing both powers and roots. The expression $a^{\frac{m}{n}}$ can be interpreted as taking the nth root of 'a' and then raising it to the power of 'm'. Mathematically, this can be written as:

$a^{\frac{m}{n}} = (\sqrt[n]{a})^m$

Where:

• 'a' is the base.
• 'm' is the numerator, representing the power.
• 'n' is the denominator, representing the root.

This fundamental understanding is critical for simplifying expressions involving fractional exponents. The denominator of the fraction indicates the type of root to be taken (square root, cube root, etc.), while the numerator indicates the power to which the result is raised. Applying this principle to our problem, $144^{\frac{3}{2}}$, we can see that we need to find the square root of 144 and then raise the result to the power of 3.

Breaking Down 144^(3/2)

Now, let's apply this knowledge to the specific expression $144^{\frac{3}{2}}$. Following the rule we established, we can rewrite this expression as:

$144^{\frac{3}{2}} = (\sqrt{144})^3$

The next step is to evaluate the square root of 144. Many will recognize that 144 is a perfect square, meaning it is the result of squaring an integer. Specifically:

$\sqrt{144} = 12$

This is a crucial step, as it simplifies the expression significantly. Now we have:

$(12)^3$

This means we need to raise 12 to the power of 3, or multiply 12 by itself three times:

$12^3 = 12 * 12 * 12$

Calculating 12^3

To calculate 12 cubed, we can perform the multiplication step by step:

$12 * 12 = 144$

Then, we multiply the result by 12 again:

$144 * 12 = 1728$

Therefore:

$12^3 = 1728$

Going back to our original expression, we have now found that:

$144^{\frac{3}{2}} = (\sqrt{144})^3 = 12^3 = 1728$

This step-by-step breakdown illustrates the power of understanding fractional exponents. By breaking down the expression into smaller, more manageable parts, we were able to arrive at the solution without relying on complex calculations or guesswork.

Examining the Answer Choices

Now that we have calculated the value of $144^{\frac{3}{2}}$ to be 1728, let's examine the answer choices provided to identify the correct one.

The answer choices were:

A. 216 B. 1,728 C. $\sqrt[3]{12}$ D. $\sqrt[3]{72}$

Comparing our calculated result (1728) with the answer choices, we can clearly see that option B, 1,728, is the correct answer. The other options are incorrect:

• Option A (216) is the result of calculating $6^3$, which is not relevant to our problem.
• Options C ($\sqrt[3]{12}$) and D ($\sqrt[3]{72}$) involve cube roots, while our calculation involved a square root and a cube, making them incorrect.

This comparison reinforces the importance of careful calculation and understanding the underlying mathematical principles. By systematically working through the problem and verifying our answer against the choices, we can confidently arrive at the correct solution.

Common Mistakes to Avoid

When dealing with fractional exponents, several common mistakes can lead to incorrect answers. Recognizing these pitfalls can help you avoid them and improve your problem-solving accuracy.

1. Misinterpreting the Fractional Exponent: One common error is to misinterpret the meaning of the fractional exponent. It's crucial to remember that the denominator represents the root, and the numerator represents the power. For example, $a^{\frac{m}{n}}$ means taking the nth root of 'a' and then raising it to the power of 'm', not the other way around. Always double-check your interpretation to avoid this mistake.

2. Incorrectly Calculating Roots: Another frequent mistake involves incorrectly calculating the roots. For example, confusing the square root of 144 with another number. It's helpful to memorize common square roots, cube roots, and other powers to speed up calculations and reduce errors. If you're unsure, take the time to verify your calculations using a calculator or by manually finding the root.

3. Order of Operations: Failing to follow the correct order of operations (PEMDAS/BODMAS) can also lead to errors. In the case of $144^{\frac{3}{2}}$, it's important to first calculate the square root of 144 and then raise the result to the power of 3. Reversing the order would lead to an incorrect answer.

4. Simplifying Prematurely: Sometimes, students try to simplify the expression prematurely, which can lead to errors. It's generally best to break down the problem into smaller, more manageable steps, as we did in this article. This allows you to focus on each operation individually and reduces the chance of making mistakes.

By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy when working with fractional exponents.

Practice Problems

To solidify your understanding of fractional exponents, it's essential to practice solving various problems. Here are a few practice problems similar to the one we discussed:

1. Evaluate $8^{\frac{2}{3}}$
2. Simplify $25^{\frac{3}{2}}$
3. Calculate $64^{\frac{1}{2}}$
4. Find the value of $27^{\frac{4}{3}}$

Working through these problems will help you develop confidence in your ability to handle fractional exponents. Remember to break down each problem into smaller steps, carefully applying the rules of exponents and roots. Check your answers and review your work to identify any areas where you may need further practice.

Conclusion

In this article, we have demystified fractional exponents and demonstrated how to solve expressions like $144^{\frac{3}{2}}$. By understanding the fundamental principles of exponents and roots, we were able to break down the problem into manageable steps and arrive at the correct solution (1728). We also discussed common mistakes to avoid and provided practice problems to help you further develop your skills. Mastering fractional exponents is crucial for success in algebra and beyond. With consistent practice and a solid understanding of the concepts, you can confidently tackle any problem involving fractional exponents.

Remember, mathematics is a journey of discovery. Embrace the challenges, learn from your mistakes, and never stop exploring the fascinating world of numbers and equations. Fractional exponents are just one piece of the puzzle, but they are an essential piece that will unlock many more mathematical concepts in your future studies.

Fractional exponents, 144^(3/2), square root, cube, exponent rules, simplifying expressions, mathematics, algebra, problem-solving, common mistakes, practice problems

Solving Fractional Exponents: A Step-by-Step Guide to 144^(3/2)