Simplifying Radical Expressions: Math Problem Solutions

Hey math enthusiasts! Let's dive into some interesting problems involving simplifying radical expressions. We'll break down each question step-by-step, making sure you understand the concepts. Get ready to sharpen those math skills! We'll solve the problems provided and then discuss a bit about the underlying principles. Let's get started, shall we?

Question 1: Finding the Simplest Form of a Cube Root

Alright guys, the first question is all about simplifying a cube root. The question asks us to find the simplest form of $\sqrt[3]{p^2q}$. This means we're looking for an equivalent expression where the radicals are reduced as much as possible. It's like unwrapping a present to see what's inside! Let's look at the given options.

Understanding Cube Roots

Before we jump into the options, let's quickly recap what a cube root is. A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because $2 \times 2 \times 2 = 8$. Now, let's apply this to our problem. We have $\sqrt[3]{p^2q}$, which means we're looking for an expression that is equivalent. Remember, the key is to express everything in terms of fractional exponents. This makes the simplification much easier. Let's break this down and see how we can tackle it.

Analyzing the Options

• Option A: $p^{\frac{2}{3}}q^{\frac{1}{3}}$ This is a strong contender! Remember that $\sqrt[3]{p^2}$ is the same as $p^{\frac{2}{3}}$ and $\sqrt[3]{q}$ is the same as $q^{\frac{1}{3}}$. Since we have $\sqrt[3]{p^2q}$, we can rewrite it as $\sqrt[3]{p^2} \times \sqrt[3]{q}$, which perfectly aligns with $p^{\frac{2}{3}}q^{\frac{1}{3}}$. So, this looks like our answer!
• Option B: $p^{\frac{3}{8}}q^{\frac{1}{8}}$ This option doesn't match our original expression. It seems to have an incorrect exponent and therefore, cannot be the correct answer.
• Option C: $p^{\frac{1}{4}}q^{\frac{1}{4}}$ Again, the exponents don't match. The original expression doesn't translate into these fractional exponents, so we can easily eliminate this option as well.
• Option D: $p^{\frac{1}{3}}q^2$ The exponents don't match, and the base is incorrect. This is not equivalent to our original expression.
• Option E: $p^3q^1$ This is definitely not the correct answer. The entire structure of this expression doesn't reflect a cube root. Also, the original variables don't carry this value.

So, the correct answer is A: $p^{\frac{2}{3}}q^{\frac{1}{3}}$!

Question 2: Solving Exponential Equations

Now, let's move on to the second problem. It's about solving an exponential equation, which is a classic math challenge. Exponential equations involve variables in the exponents. Our goal is to find the value of x that satisfies the equation. Here's the equation: $\frac{2^{x+1} \times 2^{x-1}}{16} = 8$. This requires us to remember some exponent rules. Let's do it together!

Understanding Exponential Rules

Before we start, let's quickly review a few key rules of exponents. First, when you multiply two terms with the same base, you add the exponents: $a^m \times a^n = a^{m+n}$. Second, any number to the power of 1 is itself. Also, remember that we can express any number as a power of a prime number to make it easier to solve. For example, $16 = 2^4$ and $8 = 2^3$. Knowing these rules will be key to solving our problem. Ready to use these rules?

Simplifying the Equation

Let's start by simplifying the equation. We have $\frac{2^{x+1} \times 2^{x-1}}{16} = 8$.

1. Simplify the numerator: Using the rule $a^m \times a^n = a^{m+n}$, we can simplify $2^{x+1} \times 2^{x-1}$ to $2^{(x+1) + (x-1)} = 2^{2x}$.
2. Rewrite the equation: Now, our equation becomes $\frac{2^{2x}}{16} = 8$.
3. Express everything as powers of 2: We know that $16 = 2^4$ and $8 = 2^3$. So, our equation becomes $\frac{2^{2x}}{2^4} = 2^3$.
4. Simplify the fraction: When you divide two terms with the same base, you subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$. Thus, $\frac{2^{2x}}{2^4} = 2^{2x-4}$.
5. Rewrite the equation again: Now we have $2^{2x-4} = 2^3$.
6. Equate the exponents: Since the bases are the same, we can equate the exponents: $2x - 4 = 3$.
7. Solve for x: Add 4 to both sides: $2x = 7$. Then, divide by 2: $x = \frac{7}{2}$ or $x = 3.5$. So, this is our solution to the equation!

Therefore, the value of x that satisfies the equation is 3.5!

Key Takeaways and Tips

• Master Exponent Rules: Knowing the rules of exponents is critical for solving these types of problems. Make sure you're comfortable with adding, subtracting, multiplying, and dividing exponents.
• Practice Makes Perfect: The more problems you solve, the better you'll become at recognizing patterns and applying the rules.
• Break It Down: When faced with a complex problem, break it down into smaller, more manageable steps. This makes the problem less daunting.
• Convert to Like Bases: Often, the key to simplifying exponential equations is to rewrite all terms with the same base.

Keep up the great work and never stop learning!