Simplify Math: Find The Value Of The Expression

Hey math enthusiasts! Today, we're diving into a fun algebra problem: finding the value of a given expression. Specifically, we'll tackle this: $\frac{(5^{\frac{4}{7}} \cdot 11^{\frac{3}{7}})^{21}}{55^{12}}$. Don't worry if it looks a bit intimidating at first glance; we'll break it down step-by-step to make it super easy to understand. Ready to flex those math muscles? Let's get started!

Deconstructing the Expression: A Step-by-Step Guide

Alright, guys, let's get our hands dirty and dissect this expression piece by piece. Our main goal here is to simplify the expression to its most basic form so that we can easily calculate its value. We will use a bunch of math rules in the process, so let's get to work!

First, let's focus on the numerator: $(5^{\frac{4}{7}} \cdot 11^{\frac{3}{7}})^{21}$. Remember the rule that states that $(a \cdot b)^c = a^c \cdot b^c$. Applying this, we get: $(5^{\frac{4}{7}})^{21} \cdot (11^{\frac{3}{7}})^{21}$. Now, another handy rule comes into play: $(a^b)^c = a^{b \cdot c}$. So, applying this rule to our expression: $5^{\frac{4}{7} \cdot 21} \cdot 11^{\frac{3}{7} \cdot 21}$. Simplifying the exponents, we get: $5^{12} \cdot 11^9$.

Next, let's look at the denominator: $55^{12}$. We can rewrite 55 as $5 \cdot 11$. So, the denominator becomes: $(5 \cdot 11)^{12}$. Using the rule $(a \cdot b)^c = a^c \cdot b^c$ again, we get: $5^{12} \cdot 11^{12}$.

Now that we've simplified both the numerator and the denominator, our expression looks like this: $\frac{5^{12} \cdot 11^9}{5^{12} \cdot 11^{12}}$.

Mastering Exponents: Simplifying the Expression Further

Now that we have simplified the numerator and denominator, it's time to further simplify the expression using exponent rules. This is where the magic happens, guys! Remember the rule: $\frac{a^b}{a^c} = a^{b-c}$. Let's apply this to our expression. For the powers of 5, we have $\frac{5^{12}}{5^{12}} = 5^{12-12} = 5^0$. And, for the powers of 11, we have $\frac{11^9}{11^{12}} = 11^{9-12} = 11^{-3}$.

So, our expression now simplifies to: $5^0 \cdot 11^{-3}$. Anything to the power of 0 is 1, right? So, $5^0 = 1$. This leaves us with: $1 \cdot 11^{-3}$. Remember that $a^{-b} = \frac{1}{a^b}$. Applying this rule, $11^{-3} = \frac{1}{11^3}$.

Finally, we have our simplified expression: $\frac{1}{11^3}$. To get the final answer, we just need to calculate $11^3$, which is $11 \cdot 11 \cdot 11 = 1331$. Thus, the value of the expression is $\frac{1}{1331}$. And there you have it! We've successfully simplified and found the value of the expression.

Core Concepts and Rules Used

Alright, let's recap the main mathematical concepts and rules we used to solve this problem. Understanding these concepts is key to tackling similar problems in the future. We'll break it down into easy-to-digest chunks.

Rules of Exponents

• $(a \cdot b)^c = a^c \cdot b^c$: This rule allows us to distribute an exponent over a product of terms. We used this to simplify both the numerator and the denominator by separating the powers of 5 and 11.
• $(a^b)^c = a^{b \cdot c}$: This rule helps when you have a power raised to another power. We used this to simplify the terms in the numerator.
• $\frac{a^b}{a^c} = a^{b-c}$: This rule is crucial for dividing exponential terms with the same base. We used this to simplify the expression by combining the powers of 5 and the powers of 11.
• $a^{-b} = \frac{1}{a^b}$: This rule defines negative exponents. We used it to convert $11^{-3}$ into a fraction.
• $a^0 = 1$: Any non-zero number raised to the power of 0 equals 1. We used this to simplify $5^0$.

Order of Operations (PEMDAS/BODMAS)

While not explicitly shown, we implicitly followed the order of operations (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) to ensure we performed the calculations in the correct sequence. This helps us in achieving consistent and accurate results.

Prime Factorization

In this specific problem, we didn't heavily rely on prime factorization, but it's a useful concept. Understanding prime factors (like 5 and 11) can help in breaking down numbers into simpler components, which is crucial in more complex simplification problems.

Important Considerations

• Base Understanding: The most fundamental aspect is a clear understanding of what exponents mean and how they interact with other mathematical operations.
• Practice is Key: The more you practice, the more familiar these rules will become. Work through different examples to solidify your understanding.
• Attention to Detail: Be very careful with the signs and the order of operations. A small mistake can lead to a completely wrong answer!

Practice Makes Perfect: Try These Similar Problems

Hey guys, now it's your turn to shine! Let's put those newly acquired skills to the test with some practice problems. Remember, the more you practice, the better you'll get at solving these kinds of algebra problems. I've got some similar expressions for you to work on. Give them a shot, and see if you can apply the same strategies we used earlier. Don't be afraid to make mistakes; that's how we learn and grow!

Practice Problem 1

Find the value of: $\frac{(2^{\frac{3}{4}} \cdot 7^{\frac{1}{4}})^{8}}{14^{3}}$

Practice Problem 2

Simplify: $\frac{3^{5} \cdot 9^{-2}}{3^{2}}$

Practice Problem 3

Evaluate: $\frac{(4^{\frac{1}{2}} \cdot 5^{\frac{1}{3}})^{6}}{20^{2}}$

Tips for Solving

• Break it down: Start by simplifying the numerator and denominator separately.
• Use exponent rules: Apply the exponent rules we discussed earlier.
• Look for common bases: Try to express all terms with the same base.
• Double-check: Always check your work to avoid careless errors.

Final Thoughts: Mastering Exponents

There you have it, guys! We've conquered the expression and delved deep into the fascinating world of exponents. By breaking down the problem step-by-step, we've shown how to simplify complex expressions using fundamental exponent rules. Remember, the key is to understand the rules and to practice regularly. Don't be discouraged if you find it tricky at first; the more you work with these concepts, the more natural they'll become.

So, keep practicing, exploring, and challenging yourselves with new math problems. Math is a journey, not a destination, so enjoy the process and celebrate your successes along the way. Keep learning, keep growing, and never stop exploring the amazing world of mathematics! I hope you enjoyed the lesson, and happy calculating!