Simplifying Exponential Expressions A Step-by-Step Guide To 64^(2/3)

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In the realm of mathematics, simplifying expressions involving exponents and radicals is a fundamental skill. One such expression is $64^{\frac{2}{3}}$, which combines the concepts of fractional exponents and powers. To effectively tackle this problem, we will delve into the underlying principles of exponents and radicals, break down the expression step-by-step, and provide a clear, comprehensive solution. This exploration will not only illuminate the simplification process but also reinforce your understanding of mathematical operations involving exponents.

Decoding Fractional Exponents

At the heart of simplifying $64^{\frac{2}{3}}$ lies the understanding of fractional exponents. A fractional exponent, like $\frac{2}{3}$, signifies both a root and a power. The denominator of the fraction (in this case, 3) indicates the root to be taken, while the numerator (in this case, 2) indicates the power to which the result should be raised. Therefore, $64^{\frac{2}{3}}$ can be interpreted as the cube root of 64, squared. This understanding is crucial as it allows us to transform the fractional exponent into a combination of radicals and powers, making the expression easier to handle. To further solidify this concept, let's consider a general form: $a^{\frac{m}{n}}$ can be expressed as $(\sqrt[n]{a})^m$ or $\sqrt[n]{a^m}$. This equivalence is the key to unlocking the simplification of expressions with fractional exponents.

Step-by-Step Simplification of $64^{\frac{2}{3}}$

Now that we have a grasp of fractional exponents, let's apply this knowledge to simplify $64^{\frac{2}{3}}$.

Step 1: Rewrite the Expression

Using the principle we discussed, we can rewrite $64^{\frac{2}{3}}$ as $(\sqrt[3]{64})^2$. This transformation separates the fractional exponent into a cube root and a square, making it more manageable.

Step 2: Calculate the Cube Root

Next, we need to find the cube root of 64. The cube root of a number is a value that, when multiplied by itself three times, equals the original number. In this case, we are looking for a number $x$ such that $x * x * x = 64$. By recognizing that $4 * 4 * 4 = 64$, we can determine that the cube root of 64 is 4. Therefore, $\sqrt[3]{64} = 4$.

Step 3: Square the Result

Now that we have the cube root of 64, which is 4, we need to square this result. Squaring a number means multiplying it by itself. So, we calculate $4^2$, which is $4 * 4 = 16$.

Final Answer

Putting it all together, we have $64^{\frac{2}{3}} = (\sqrt[3]{64})^2 = 4^2 = 16$. Therefore, the simplified form of $64^{\frac{2}{3}}$ is 16.

Alternative Approach: Power First, Then Root

While we simplified $64^{\frac{2}{3}}$ by taking the cube root first and then squaring, there's an alternative approach: squaring first and then taking the cube root. This approach, though mathematically equivalent, can sometimes be more challenging depending on the numbers involved. Let's explore this alternative method to provide a more comprehensive understanding.

Step 1: Rewrite the Expression

As before, we start by rewriting $64^{\frac{2}{3}}$, but this time we express it as $\sqrt[3]{64^2}$. This form emphasizes squaring 64 before taking the cube root.

Step 2: Calculate the Power

We need to calculate $64^2$, which is $64 * 64$. This multiplication yields 4096. So, we have $\sqrt[3]{4096}$.

Step 3: Calculate the Cube Root

Now, we need to find the cube root of 4096. This means finding a number that, when multiplied by itself three times, equals 4096. This step can be more challenging than finding the cube root of 64 directly. However, through prime factorization or recognition of powers, we can determine that the cube root of 4096 is 16. Therefore, $\sqrt[3]{4096} = 16$.

Final Answer

As we see, even with this alternative approach, we arrive at the same answer: $64^{\frac{2}{3}} = \sqrt[3]{64^2} = \sqrt[3]{4096} = 16$. This demonstrates that the order of operations (root then power or power then root) does not affect the final result, although one method might be computationally simpler than the other in certain cases.

Key Concepts and Properties

To master the simplification of expressions like $64^{\frac{2}{3}}$, it's essential to understand the underlying mathematical concepts and properties. Let's review some of the key principles that are fundamental to this process:

1. Fractional Exponents

As we've discussed, a fractional exponent represents both a root and a power. The expression $a^{\frac{m}{n}}$ can be interpreted as the nth root of a raised to the power of m, or $(\sqrt[n]{a})^m$. This property allows us to convert fractional exponents into a more manageable form involving radicals and powers.

2. Laws of Exponents

The laws of exponents are a set of rules that govern how exponents interact with each other in various mathematical operations. Some of the key laws include:

• Product of Powers: $a^m * a^n = a^{m+n}$ (When multiplying powers with the same base, add the exponents).
• Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$ (When dividing powers with the same base, subtract the exponents).
• Power of a Power: $(a^m)^n = a^{m*n}$ (When raising a power to another power, multiply the exponents).
• Power of a Product: $(ab)^n = a^n * b^n$ (The power of a product is the product of the powers).
• Power of a Quotient: $(\frac{a}{b})^n = \frac{a^n}{b^n}$ (The power of a quotient is the quotient of the powers).

3. Radicals and Roots

A radical, denoted by the symbol $\sqrt[n]{}$, represents the nth root of a number. The index n indicates the type of root (e.g., square root, cube root). Understanding how to find roots of numbers, especially perfect squares, cubes, and higher powers, is crucial for simplifying expressions involving radicals and fractional exponents.

4. Prime Factorization

Prime factorization is the process of breaking down a number into its prime factors (numbers that are only divisible by 1 and themselves). This technique can be particularly useful when finding roots of larger numbers. By expressing a number as a product of its prime factors, we can identify perfect nth powers and simplify the radical expression.

Common Mistakes to Avoid

When simplifying expressions with fractional exponents, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

1. Incorrectly Interpreting Fractional Exponents

A frequent error is misinterpreting the meaning of the numerator and denominator in a fractional exponent. Remember, the denominator indicates the root, and the numerator indicates the power. Failing to distinguish between these can lead to incorrect simplification.

2. Applying the Power Before the Root When It's More Difficult

While mathematically correct, squaring a number before taking the root can sometimes result in larger numbers that are harder to work with. It's often simpler to take the root first, especially if the number is a perfect square, cube, or higher power.

3. Forgetting the Laws of Exponents

The laws of exponents are essential tools for simplifying expressions. Neglecting to apply these rules correctly can lead to errors. Make sure you have a solid understanding of these laws and when to apply them.

4. Arithmetic Errors

Simple arithmetic mistakes, such as miscalculating a square or a cube root, can derail the entire simplification process. Double-check your calculations to ensure accuracy.

5. Not Simplifying Completely

Sometimes, students may perform the initial steps correctly but fail to simplify the expression completely. Always ensure that your final answer is in its simplest form.

Practice Problems

To reinforce your understanding of simplifying expressions with fractional exponents, let's work through some practice problems:

1. Simplify $8^{\frac{2}{3}}$
2. Simplify $27^{\frac{4}{3}}$
3. Simplify $16^{\frac{3}{4}}$
4. Simplify $125^{\frac{2}{3}}$
5. Simplify $32^{\frac{2}{5}}$

Solutions:

1. $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$
2. $27^{\frac{4}{3}} = (\sqrt[3]{27})^4 = 3^4 = 81$
3. $16^{\frac{3}{4}} = (\sqrt[4]{16})^3 = 2^3 = 8$
4. $125^{\frac{2}{3}} = (\sqrt[3]{125})^2 = 5^2 = 25$
5. $32^{\frac{2}{5}} = (\sqrt[5]{32})^2 = 2^2 = 4$

By working through these examples, you can solidify your understanding of the simplification process and build confidence in your ability to tackle similar problems.

Conclusion

Simplifying expressions like $64^{\frac{2}{3}}$ requires a solid grasp of fractional exponents, radicals, and the laws of exponents. By breaking down the problem into manageable steps, understanding the underlying principles, and avoiding common mistakes, you can confidently simplify these types of expressions. Remember to practice regularly and review the key concepts to solidify your understanding. Mastering these skills will not only enhance your mathematical abilities but also provide a strong foundation for more advanced topics in algebra and calculus. The ability to manipulate exponents and radicals is a cornerstone of mathematical fluency, and with practice, you can achieve proficiency in this area.