Solve For M: P^m = 1/(p * Cube_root(p^2))

Oct 27, 2025 by SLV Team 42 views

Hey guys! Let's dive into solving this equation where we need to find the value of 'm' given the equation: $p^m=\frac{1}{p \times \sqrt[3]{p^2}}$ . This is a fun problem involving exponents and radicals, so let's break it down step by step to make sure we understand each part clearly. Our main goal is to express both sides of the equation with the same base 'p', which will allow us to equate the exponents and solve for 'm'.

Step 1: Rewrite the Radical as an Exponent

The first thing we need to do is to convert the cube root of $p^2$ into an exponential form. Remember that a radical can be written as a fractional exponent. Specifically, the $n^{th}$ root of $a^m$ can be written as $a^{\frac{m}{n}}$ . In our case, we have the cube root of $p^2$ , which can be written as $p^{\frac{2}{3}}$ . So, our equation now looks like this:

$p^m = \frac{1}{p \times p^{\frac{2}{3}}}$

This transformation helps us to combine the terms in the denominator more easily, as they are now both expressed as powers of 'p'. It's all about making the equation easier to manipulate!

Step 2: Combine the Terms in the Denominator

Now that we have both terms in the denominator as powers of 'p', we can combine them. When you multiply terms with the same base, you add their exponents. So, we have:

$p \times p^{\frac{2}{3}} = p^{1 + \frac{2}{3}}$

To add the exponents, we need to find a common denominator. In this case, we can write 1 as $\frac{3}{3}$ , so we have:

$1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}$

Thus, the denominator simplifies to $p^{\frac{5}{3}}$ . Our equation now looks like this:

$p^m = \frac{1}{p^{\frac{5}{3}}}$

Step 3: Rewrite the Right Side with a Negative Exponent

Next, we want to get rid of the fraction. Remember that $\frac{1}{a^n}$ is the same as $a^{-n}$ . So, we can rewrite the right side of the equation as:

$\frac{1}{p^{\frac{5}{3}}} = p^{-\frac{5}{3}}$

Now, our equation looks like this:

$p^m = p^{-\frac{5}{3}}$

Step 4: Equate the Exponents

Now that we have both sides of the equation expressed as powers of 'p', we can simply equate the exponents. This means that:

$m = -\frac{5}{3}$

So, the value of 'm' that satisfies the equation is $-\frac{5}{3}$ .

Final Answer

Therefore, the value of m is: m = -5/3

In summary, we solved for 'm' by first converting the radical to an exponent, combining terms with the same base, rewriting the fraction using a negative exponent, and then equating the exponents. I hope this detailed explanation helps you understand each step of the solution! Let me know if you have any more questions. This process demonstrates the power of manipulating exponents and radicals to solve equations efficiently. Keep practicing, and you'll become a pro at these types of problems! Remember, the key is to break down the problem into smaller, manageable steps. Each step builds upon the previous one, leading you to the final solution. Happy problem-solving!

Let's try to make it even more detailed.

Understanding the Basics

Before we even start, let's make sure we're all on the same page with some fundamental concepts. Exponents and radicals are two sides of the same coin. An exponent tells you how many times to multiply a number by itself, while a radical (like a square root or cube root) asks the question, "What number, when multiplied by itself a certain number of times, gives you this number?"

The relationship between exponents and radicals is crucial here. A radical can always be expressed as a fractional exponent. For example:

$\sqrt{x} = x^{\frac{1}{2}}$ (square root)
$\sqrt[3]{x} = x^{\frac{1}{3}}$ (cube root)
$\sqrt[n]{x} = x^{\frac{1}{n}}$ ( $n^{th}$ root)

And, more generally:

$\sqrt[n]{x^m} = x^{\frac{m}{n}}$

This is exactly what we used when we converted $\sqrt[3]{p^2}$ to $p^{\frac{2}{3}}$ in our problem.

Why This Matters

Why bother converting radicals to exponents? Because exponents are often easier to manipulate in equations. We have well-defined rules for how exponents behave when multiplying, dividing, and raising to further powers. Radicals, on the other hand, can be a bit clunkier to work with directly.

Breaking Down the Original Equation

Our original equation is:

$p^m = \frac{1}{p \times \sqrt[3]{p^2}}$

Notice how the right side is a bit of a mess? We've got a 'p' multiplied by a cube root of $p^2$ , and the whole thing is in the denominator of a fraction. That's why our first steps focus on cleaning up the right side.

The Power of Rewriting

Mathematics is all about rewriting things in different forms to make them easier to understand and solve. That's what we're doing at each step:

Rewriting the radical: $\sqrt[3]{p^2}$ becomes $p^{\frac{2}{3}}$
Combining terms in the denominator: $p \times p^{\frac{2}{3}}$ becomes $p^{\frac{5}{3}}$
Getting rid of the fraction: $\frac{1}{p^{\frac{5}{3}}}$ becomes $p^{-\frac{5}{3}}$

Each of these rewrites makes the equation simpler and brings us closer to isolating 'm'.

Why Negative Exponents?

A negative exponent indicates the reciprocal of the base raised to the positive version of that exponent. In other words:

$x^{-n} = \frac{1}{x^n}$

This is a super useful trick for getting rid of fractions in equations. In our case, we used it to move $p^{\frac{5}{3}}$ from the denominator to the numerator, but with a negative sign in the exponent.

Equating Exponents: The Final Step

Once we have both sides of the equation in the form $p^{\text{something}} = p^{\text{something else}}$ , we can equate the "something" and "something else." This is because if two powers with the same base are equal, then their exponents must be equal. In our case, we ended up with:

$p^m = p^{-\frac{5}{3}}$

Therefore, $m = -\frac{5}{3}$ .

Checking Your Work

It's always a good idea to check your answer by plugging it back into the original equation. Let's do that:

Original equation: $p^m = \frac{1}{p \times \sqrt[3]{p^2}}$

Substitute $m = -\frac{5}{3}$ :

$p^{-\frac{5}{3}} = \frac{1}{p \times \sqrt[3]{p^2}}$

We already know that the right side simplifies to $p^{-\frac{5}{3}}$ , so the equation holds true! This confirms that our answer is correct.

Common Mistakes to Avoid

Forgetting the rules of exponents: Make sure you know how to add, subtract, multiply, and divide exponents correctly.
Incorrectly converting radicals to exponents: Remember that $\sqrt[n]{x^m} = x^{\frac{m}{n}}$ .
Messing up negative signs: Be careful when dealing with negative exponents and make sure you understand what they mean.
Not checking your work: Always plug your answer back into the original equation to make sure it's correct.

Tips for Mastering These Types of Problems

Practice, practice, practice: The more you work with exponents and radicals, the more comfortable you'll become with them.
Review the rules of exponents: Make sure you have a solid understanding of the basic rules.
Break down complex problems into smaller steps: Don't try to do everything at once. Focus on simplifying one part of the equation at a time.
Check your work: Always double-check your answer to make sure it's correct.
Don't be afraid to ask for help: If you're stuck, don't hesitate to ask a teacher, tutor, or friend for help.

Final Thoughts

Solving equations with exponents and radicals can seem daunting at first, but with a solid understanding of the basic principles and a bit of practice, you can master them. Remember to break down complex problems into smaller, more manageable steps, and always check your work to make sure your answer is correct. Keep practicing, and you'll become a pro at these types of problems in no time!