Simplify $\frac{\sqrt[3]{x^2} \cdot \sqrt{x}}{\sqrt[5]{x^3}}$ For $x>0$

Hey guys! Today, we're diving deep into the awesome world of simplifying expressions, specifically tackling one that looks a little intimidating at first glance: $\frac{\sqrt[3]{x^2} \cdot \sqrt{x}}{\sqrt[5]{x^3}}$. Don't sweat it, though! By the end of this article, you'll be a pro at simplifying this bad boy and ready to conquer any similar math challenges. We'll break it down step-by-step, showing you how to transform this radical beast into two super clean, equivalent forms: one with a radical and one with a rational exponent. So, grab your calculators, a comfy seat, and let's get this math party started!

Understanding the Building Blocks: Radicals and Exponents

Before we jump into the simplification, let's make sure we're all on the same page with the fundamental concepts. You've probably seen radicals (like square roots, cube roots, etc.) and exponents (those little numbers floating above the base) before. They're basically two sides of the same coin, representing powers and roots. The key to simplifying our expression lies in understanding how to convert between these two forms. A radical like $\sqrt[n]{a^m}$ can be rewritten as $a^{m/n}$. This is a super powerful tool, guys, because it allows us to use the rules of exponents, which are often much easier to manipulate, to simplify expressions involving radicals. So, when you see a radical, think 'fractional exponent'! For our problem, we have $\sqrt[3]{x^2}$, which is equivalent to $x^{2/3}$; $\sqrt{x}$ (which is a square root, so n=2), equivalent to $x^{1/2}$; and $\sqrt[5]{x^3}$, equivalent to $x^{3/5}$. See? We're already halfway there just by translating!

Step 1: Convert All Radicals to Rational Exponents

Alright, team, let's kick things off by translating our given expression into the language of rational exponents. Our original expression is $\frac{\sqrt[3]{x^2} \cdot \sqrt{x}}{\sqrt[5]{x^3}}$.

• The term $\sqrt[3]{x^2}$ becomes $x^{2/3}$.
• The term $\sqrt{x}$ is the same as $\sqrt[2]{x^1}$, which becomes $x^{1/2}$.
• The term $\sqrt[5]{x^3}$ becomes $x^{3/5}$.

Now, substitute these back into the original expression:

$\frac{x^{2/3} \cdot x^{1/2}}{x^{3/5}}$

This looks way less scary, right? We've successfully converted all the radicals into their exponential forms. This is a crucial first step because now we can apply the handy-dandy rules of exponents to combine these terms. Remember, when you're dealing with expressions like this, the goal is always to simplify them as much as possible, and converting to exponents is usually the golden ticket to achieving that.

Step 2: Simplify the Numerator Using the Product of Powers Rule

Next up, guys, we need to tackle the numerator: $x^{2/3} \cdot x^{1/2}$. When you multiply terms with the same base, you add their exponents. This is the Product of Powers Rule, a fundamental concept in exponent arithmetic. So, we need to add $2/3$ and $1/2$:

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

To add fractions, we need a common denominator. The least common denominator for 3 and 2 is 6. So, we convert the fractions:

• $\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$
• $\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

Now, add them:

$\frac{4}{6} + \frac{3}{6} = \frac{4+3}{6} = \frac{7}{6}$

So, the numerator simplifies to $x^{7/6}$. Our expression now looks like this:

$\frac{x^{7/6}}{x^{3/5}}$

See how we're systematically simplifying? By applying one rule at a time, we're breaking down a complex problem into manageable steps. Keep that focus, and you'll master this!

Step 3: Simplify the Fraction Using the Quotient of Powers Rule

We're on a roll, team! Now we have a fraction with the same base in the numerator and denominator: $\frac{x^{7/6}}{x^{3/5}}$. When you divide terms with the same base, you subtract the exponent in the denominator from the exponent in the numerator. This is the Quotient of Powers Rule. So, we need to subtract $3/5$ from $7/6$:

$\frac{7}{6} - \frac{3}{5}$

Again, we need a common denominator to subtract these fractions. The least common denominator for 6 and 5 is 30. Let's convert:

• $\frac{7}{6} = \frac{7 \times 5}{6 \times 5} = \frac{35}{30}$
• $\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}$

Now, perform the subtraction:

$\frac{35}{30} - \frac{18}{30} = \frac{35-18}{30} = \frac{17}{30}$

So, the entire expression simplifies to $x^{17/30}$. This is our answer in the rational exponent form!

Step 4: Convert Back to Radical Form

We've nailed the rational exponent form, but the question asked for two equivalent forms: one with a rational exponent and one as a radical. Luckily, converting back is just as easy as converting over. Remember our golden rule: $a^{m/n} = \sqrt[n]{a^m}$.

Our simplified expression is $x^{17/30}$. Here, the numerator of the exponent (17) is our 'm' (the power of the base), and the denominator of the exponent (30) is our 'n' (the index of the root).

So, $x^{17/30}$ can be written as:

$\sqrt[30]{x^{17}}$

And there you have it! Our expression simplified into its radical form. This is the second equivalent form we were looking for.

Final Answer: Two Equivalent Forms

To recap, guys, we started with the expression $\frac{\sqrt[3]{x^2} \cdot \sqrt{x}}{\sqrt[5]{x^3}}$ and, by applying the rules of exponents after converting radicals to rational exponents, we simplified it.

1. Rational Exponent Form:

$x^{17/30}$

2. Radical Form:

$\sqrt[30]{x^{17}}$

Both of these forms are completely equivalent and represent the simplest version of our original, more complex expression. Remember, the key takeaways are:

• Convert radicals to rational exponents: $\sqrt[n]{a^m} = a^{m/n}$
• Product of Powers Rule: $a^m \cdot a^n = a^{m+n}$
• Quotient of Powers Rule: $\frac{a^m}{a^n} = a^{m-n}$

By mastering these fundamental rules and practicing them, you'll find that simplifying complex algebraic expressions becomes much less daunting and, dare I say, even a little bit fun! Keep practicing, and you'll be a simplification wizard in no time. Stay curious, and happy calculating!