Simplify The Expression: A Math Problem

Let's break down how to simplify the expression:

${\frac{\left(\frac{1}{27^{\frac{1}{3}}} \cdot \frac{1}{16^{\frac{1}{4}}}\right)^2}{8^{\frac{2}{3}} \cdot 125^{-\frac{1}{3}}} = ?}$

We'll go through each part step by step, making it super easy to follow. So, grab your calculator (or your brain!) and let's dive in!

Breaking Down the Expression

Understanding the Basics

When we simplify expressions, we're essentially trying to make them as neat and manageable as possible. This often involves dealing with exponents and fractions. The key here is to remember the rules of exponents and how to manipulate fractions efficiently.

First, let's recall some fundamental exponent rules. For any number a and exponents m and n:

1. (a m ) n = a m*n
2. a -n = 1/a n

Also, remember that a fractional exponent like a^(1/n) means the nth root of a. For example, 8^(1/3) is the cube root of 8, which is 2.

Now, let’s tackle each term in our expression piece by piece to make sure we understand every single operation. This methodical approach will help avoid errors and make the entire simplification process smoother.

Simplifying Individual Terms

Okay, let's start by simplifying each term individually.

1. 27^(1/3): This is the cube root of 27. Since 3 * 3 * 3 = 27, we have 27^(1/3) = 3.
2. 16^(1/4): This is the fourth root of 16. Since 2 * 2 * 2 * 2 = 16, we have 16^(1/4) = 2.
3. 8^(2/3): This can be seen as (8^(1/3))2. The cube root of 8 is 2, so we have 2^2 = 4.
4. 125^(-1/3): This is 1 / (125^(1/3)). The cube root of 125 is 5, so we have 1 / 5.

Now that we've simplified each term, let's substitute these values back into the original expression. This will make the entire expression much easier to manage and reduce the chances of making mistakes.

Putting It All Together

Now, let's substitute the simplified values back into the original expression:

${\frac{\left(\frac{1}{3} \cdot \frac{1}{2}\right)^2}{4 \cdot \frac{1}{5}}}$

Simplify the Numerator

First, we simplify the terms inside the parentheses:

${\frac{1}{3} \cdot \frac{1}{2} = \frac{1}{6}}$

Now, square this result:

${\left(\frac{1}{6}\right)^2 = \frac{1}{36}}$

So, the numerator becomes:

${\frac{1}{36}}$

Simplify the Denominator

Next, we simplify the denominator:

${4 \cdot \frac{1}{5} = \frac{4}{5}}$

Divide Numerator by Denominator

Now, we divide the simplified numerator by the simplified denominator:

${\frac{\frac{1}{36}}{\frac{4}{5}} = \frac{1}{36} \div \frac{4}{5} = \frac{1}{36} \cdot \frac{5}{4}}$

Multiply the fractions:

${\frac{1 \cdot 5}{36 \cdot 4} = \frac{5}{144}}$

So, the simplified expression is:

${\frac{5}{144}}$

Therefore, the final simplified expression is ${\frac{5}{144}}$

Detailed Step-by-Step Solution

Let's go through the entire process step-by-step to ensure complete clarity. Guys, this way, you can follow along and double-check each operation.

Step 1: Simplify the Terms Inside the Parentheses

We start with the terms inside the parentheses:

${\frac{1}{27^{\frac{1}{3}}} \cdot \frac{1}{16^{\frac{1}{4}}}}$

We know that 27^(1/3) = 3 and 16^(1/4) = 2. So we substitute these values:

${\frac{1}{3} \cdot \frac{1}{2} = \frac{1}{6}}$

Step 2: Square the Result from Step 1

Now we square the result:

${\left(\frac{1}{6}\right)^2 = \frac{1}{36}}$

Step 3: Simplify the Terms in the Denominator

Now let's simplify the denominator:

${8^{\frac{2}{3}} \cdot 125^{-\frac{1}{3}}}$

We know that 8^(2/3) = (8^(1/3))2 = 2^2 = 4 and 125^(-1/3) = 1 / (125^(1/3)) = 1 / 5. So we substitute these values:

${4 \cdot \frac{1}{5} = \frac{4}{5}}$

Step 4: Divide the Numerator by the Denominator

Now we divide the numerator by the denominator:

${\frac{\frac{1}{36}}{\frac{4}{5}} = \frac{1}{36} \div \frac{4}{5} = \frac{1}{36} \cdot \frac{5}{4}}$

Step 5: Multiply the Fractions

Multiply the fractions to get the final result:

${\frac{1 \cdot 5}{36 \cdot 4} = \frac{5}{144}}$

Common Mistakes to Avoid

When simplifying expressions like this, there are a few common mistakes that students often make. Knowing these pitfalls can help you avoid them and ensure you get the correct answer.

Incorrectly Applying Exponent Rules

One of the most common mistakes is misapplying exponent rules. For instance, students might incorrectly multiply the bases instead of adding the exponents when multiplying terms with the same base. Always double-check which rule applies to the specific situation. For example, remember that (a m ) n = a mn *, not a^(m+n).

Miscalculating Roots

Another frequent error is miscalculating roots, especially fractional exponents. For example, confusing the cube root with the square root is a common mistake. Make sure you understand what each fractional exponent represents. Remember, a^(1/n) means the nth root of a. Always verify your roots by multiplying the result by itself the correct number of times.

Forgetting Negative Exponents

Negative exponents can also cause confusion. Remember that a negative exponent means taking the reciprocal of the base raised to the positive exponent. For example, a^-n = 1/a^n. Forgetting this rule can lead to incorrect simplifications.

Arithmetic Errors

Simple arithmetic errors, such as incorrect multiplication or division, can also lead to wrong answers. Always double-check your calculations, especially when dealing with fractions. Writing out each step clearly can help you spot these errors more easily.

Not Simplifying Completely

Finally, failing to simplify the expression completely is another common mistake. Ensure that you have reduced the expression to its simplest form. This often means combining like terms, reducing fractions, and eliminating any remaining exponents.

Conclusion

The simplified form of the expression is:

${\frac{5}{144}}$

So, by carefully breaking down each component and methodically applying the rules of exponents and fractions, we arrive at the simplified form. Remember, practice makes perfect! Keep simplifying expressions, and soon it'll become second nature. You got this!