Simplify.${\frac{54 W^6}{9 W^4} = \square}$

Understanding the Problem

When simplifying an expression, we need to reduce it to its simplest form by canceling out common factors. In this case, we have a fraction with a variable in the numerator and denominator. Our goal is to simplify the expression and find its value.

Simplifying the Expression

To simplify the expression, we can start by canceling out any common factors in the numerator and denominator. In this case, we have a common factor of $w^4$ in both the numerator and denominator.

\frac{54 w^6}{9 w^4} = \frac{6w^2}{1}

Canceling Out Common Factors

Now that we have canceled out the common factor of $w^4$, we can simplify the expression further by canceling out any other common factors. In this case, we have a common factor of 3 in both the numerator and denominator.

\frac{6w^2}{1} = \frac{2w^2}{\frac{1}{3}}

Simplifying the Fraction

Now that we have canceled out the common factor of 3, we can simplify the fraction by multiplying the numerator and denominator by 3.

\frac{2w^2}{\frac{1}{3}} = 6w^2

Final Answer

Therefore, the simplified expression is $6w^2$.

Conclusion

Simplifying an expression is an important step in solving mathematical problems. By canceling out common factors and simplifying fractions, we can reduce complex expressions to their simplest form. In this case, we simplified the expression $\frac{54 w^6}{9 w^4}$ to its simplest form, which is $6w^2$.

Real-World Applications

Simplifying expressions has many real-world applications. For example, in physics, we often need to simplify complex equations to understand the behavior of physical systems. In engineering, we use simplification techniques to design and optimize complex systems. In finance, we use simplification techniques to analyze and understand complex financial models.

Tips and Tricks

Here are some tips and tricks for simplifying expressions:

• Cancel out common factors: When simplifying an expression, look for common factors in the numerator and denominator and cancel them out.
• Simplify fractions: When simplifying a fraction, multiply the numerator and denominator by the same value to eliminate the fraction.
• Use algebraic manipulations: Use algebraic manipulations such as factoring and expanding to simplify expressions.
• Check your work: Always check your work to ensure that the simplified expression is correct.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions:

• Not canceling out common factors: Failing to cancel out common factors can lead to incorrect simplified expressions.
• Not simplifying fractions: Failing to simplify fractions can lead to complex and difficult-to-understand expressions.
• Not using algebraic manipulations: Failing to use algebraic manipulations can lead to incorrect simplified expressions.
• Not checking your work: Failing to check your work can lead to incorrect simplified expressions.

Final Thoughts

Simplifying expressions is an important step in solving mathematical problems. By canceling out common factors, simplifying fractions, and using algebraic manipulations, we can reduce complex expressions to their simplest form. Remember to check your work and avoid common mistakes to ensure that your simplified expressions are correct.

Frequently Asked Questions

We have received many questions about simplifying expressions, and we are happy to provide answers to some of the most frequently asked questions.

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to look for common factors in the numerator and denominator and cancel them out.

Q: How do I simplify a fraction?

A: To simplify a fraction, multiply the numerator and denominator by the same value to eliminate the fraction.

Q: What are some common algebraic manipulations that I can use to simplify expressions?

A: Some common algebraic manipulations that you can use to simplify expressions include factoring and expanding.

Q: How do I check my work when simplifying an expression?

A: To check your work when simplifying an expression, plug in a value for the variable and see if the expression simplifies to the correct value.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include not canceling out common factors, not simplifying fractions, and not using algebraic manipulations.

Q: Can you provide an example of how to simplify an expression using algebraic manipulations?

A: Here is an example of how to simplify an expression using algebraic manipulations:

Suppose we want to simplify the expression $\frac{x^2 + 4x + 4}{x + 2}$. We can start by factoring the numerator:

$$\frac{x^2 + 4x + 4}{x + 2} = \frac{(x + 2)^2}{x + 2}$$

Next, we can cancel out the common factor of $(x + 2)$:

$$\frac{(x + 2)^2}{x + 2} = x + 2$$

Therefore, the simplified expression is $x + 2$.

Q: Can you provide an example of how to simplify a fraction using algebraic manipulations?

A: Here is an example of how to simplify a fraction using algebraic manipulations:

Suppose we want to simplify the fraction $\frac{2x^2 + 6x + 6}{x^2 + 3x + 3}$. We can start by factoring the numerator and denominator:

$$\frac{2x^2 + 6x + 6}{x^2 + 3x + 3} = \frac{2(x + 1)(x + 3)}{(x + 1)(x + 3)}$$

Next, we can cancel out the common factor of $(x + 1)(x + 3)$:

$$\frac{2(x + 1)(x + 3)}{(x + 1)(x + 3)} = 2$$

Therefore, the simplified fraction is $2$.

Q: Can you provide an example of how to check your work when simplifying an expression?

A: Here is an example of how to check your work when simplifying an expression:

Suppose we want to simplify the expression $\frac{x^2 + 4x + 4}{x + 2}$. We can start by factoring the numerator:

$$\frac{x^2 + 4x + 4}{x + 2} = \frac{(x + 2)^2}{x + 2}$$

Next, we can cancel out the common factor of $(x + 2)$:

$$\frac{(x + 2)^2}{x + 2} = x + 2$$

To check our work, we can plug in a value for $x$ and see if the expression simplifies to the correct value. Let's say we plug in $x = 1$. Then, we get:

$$\frac{(1 + 2)^2}{1 + 2} = \frac{9}{3} = 3$$

Since $x + 2 = 1 + 2 = 3$, we can see that our simplified expression is correct.

Q: Can you provide an example of how to avoid common mistakes when simplifying expressions?

A: Here is an example of how to avoid common mistakes when simplifying expressions:

Suppose we want to simplify the expression $\frac{x^2 + 4x + 4}{x + 2}$. We can start by factoring the numerator:

$$\frac{x^2 + 4x + 4}{x + 2} = \frac{(x + 2)^2}{x + 2}$$

Next, we can cancel out the common factor of $(x + 2)$:

$$\frac{(x + 2)^2}{x + 2} = x + 2$$

However, if we don't cancel out the common factor of $(x + 2)$, we will get an incorrect simplified expression:

$$\frac{(x + 2)^2}{x + 2} = (x + 2)$$

Therefore, it's essential to cancel out common factors when simplifying expressions.

Conclusion

Simplifying expressions is an essential step in solving mathematical problems. By canceling out common factors, simplifying fractions, and using algebraic manipulations, we can reduce complex expressions to their simplest form. Remember to check your work and avoid common mistakes to ensure that your simplified expressions are correct.