Simplify: 4 72 + 11 63 − 2 28 4 \sqrt{72} + 11 \sqrt{63} - 2 \sqrt{28} 4 72 ​ + 11 63 ​ − 2 28 ​

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Introduction

Simplifying radical expressions is a crucial skill in mathematics, particularly in algebra and geometry. It involves expressing a given radical expression in its simplest form, which can be achieved by factoring the radicand and simplifying the resulting expression. In this article, we will simplify the given expression $4 \sqrt{72} + 11 \sqrt{63} - 2 \sqrt{28}$ using various techniques.

Understanding the Radicands

The given expression contains three radical terms with different radicands: $72$, $63$, and $28$. To simplify these expressions, we need to factor each radicand into its prime factors.

Factoring the Radicands

• $72 = 2^3 \times 3^2$
• $63 = 3^2 \times 7$
• $28 = 2^2 \times 7$

Simplifying the Radical Terms

Now that we have factored the radicands, we can simplify each radical term by expressing it in terms of its prime factors.

Simplifying $4 \sqrt{72}$

We can rewrite $4 \sqrt{72}$ as $4 \sqrt{2^3 \times 3^2}$. Using the property of radicals that $\sqrt{a^2} = a$, we can simplify this expression as $4 \times 2 \times \sqrt{3^2} = 8 \times 3 = 24$.

Simplifying $11 \sqrt{63}$

We can rewrite $11 \sqrt{63}$ as $11 \sqrt{3^2 \times 7}$. Using the property of radicals that $\sqrt{a^2} = a$, we can simplify this expression as $11 \times 3 \times \sqrt{7} = 33 \times \sqrt{7}$.

Simplifying $-2 \sqrt{28}$

We can rewrite $-2 \sqrt{28}$ as $-2 \sqrt{2^2 \times 7}$. Using the property of radicals that $\sqrt{a^2} = a$, we can simplify this expression as $-2 \times 2 \times \sqrt{7} = -4 \times \sqrt{7}$.

Combining the Simplified Terms

Now that we have simplified each radical term, we can combine them to obtain the final simplified expression.

Final Simplified Expression

The final simplified expression is $24 + 33 \times \sqrt{7} - 4 \times \sqrt{7}$. Combining like terms, we get $24 + 29 \times \sqrt{7}$.

Conclusion

Simplifying radical expressions is an essential skill in mathematics, and it requires a thorough understanding of the properties of radicals. By factoring the radicands and simplifying the resulting expressions, we can obtain the final simplified expression. In this article, we simplified the given expression $4 \sqrt{72} + 11 \sqrt{63} - 2 \sqrt{28}$ using various techniques and obtained the final simplified expression $24 + 29 \times \sqrt{7}$.

Additional Tips and Tricks

• Use the property of radicals that $\sqrt{a^2} = a$ to simplify radical expressions.
• Factor the radicand into its prime factors to simplify radical expressions.
• Combine like terms to simplify radical expressions.

Frequently Asked Questions

• What is the final simplified expression of $4 \sqrt{72} + 11 \sqrt{63} - 2 \sqrt{28}$?
  • The final simplified expression is $24 + 29 \times \sqrt{7}$.
• How do I simplify radical expressions?
  • To simplify radical expressions, you need to factor the radicand into its prime factors and use the property of radicals that $\sqrt{a^2} = a$ to simplify the resulting expression.

Final Thoughts

Simplifying radical expressions is a crucial skill in mathematics, and it requires a thorough understanding of the properties of radicals. By factoring the radicands and simplifying the resulting expressions, we can obtain the final simplified expression. In this article, we simplified the given expression $4 \sqrt{72} + 11 \sqrt{63} - 2 \sqrt{28}$ using various techniques and obtained the final simplified expression $24 + 29 \times \sqrt{7}$.

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Introduction

In our previous article, we simplified the given expression $4 \sqrt{72} + 11 \sqrt{63} - 2 \sqrt{28}$ using various techniques. In this article, we will provide a Q&A section to address any questions or concerns that readers may have.

Q&A

Q: What is the final simplified expression of $4 \sqrt{72} + 11 \sqrt{63} - 2 \sqrt{28}$?

A: The final simplified expression is $24 + 29 \times \sqrt{7}$.

Q: How do I simplify radical expressions?

A: To simplify radical expressions, you need to factor the radicand into its prime factors and use the property of radicals that $\sqrt{a^2} = a$ to simplify the resulting expression.

Q: What is the property of radicals that $\sqrt{a^2} = a$?

A: The property of radicals that $\sqrt{a^2} = a$ states that the square root of a perfect square is equal to the number itself. For example, $\sqrt{16} = 4$ because $4^2 = 16$.

Q: How do I factor the radicand into its prime factors?

A: To factor the radicand into its prime factors, you need to express the radicand as a product of prime numbers. For example, the radicand $72$ can be factored as $2^3 \times 3^2$.

Q: What is the difference between a perfect square and a non-perfect square?

A: A perfect square is a number that can be expressed as the square of an integer. For example, $16$ is a perfect square because $4^2 = 16$. A non-perfect square is a number that cannot be expressed as the square of an integer. For example, $7$ is a non-perfect square.

Q: How do I combine like terms to simplify radical expressions?

A: To combine like terms to simplify radical expressions, you need to add or subtract the coefficients of the like terms. For example, $2 \times \sqrt{7} + 3 \times \sqrt{7} = 5 \times \sqrt{7}$.

Q: What is the final simplified expression of $4 \sqrt{72} + 11 \sqrt{63} - 2 \sqrt{28}$ in terms of the number of decimal places?

A: The final simplified expression is $24 + 29 \times \sqrt{7}$. To express this in terms of the number of decimal places, we need to calculate the value of $\sqrt{7}$ and then multiply it by $29$. The value of $\sqrt{7}$ is approximately $2.645751311$, so the final simplified expression is approximately $24 + 29 \times 2.645751311 = 24 + 76.29999983 = 100.29999983$.

Conclusion

In this article, we provided a Q&A section to address any questions or concerns that readers may have. We covered topics such as simplifying radical expressions, factoring the radicand into its prime factors, and combining like terms to simplify radical expressions. We also provided examples and explanations to help readers understand the concepts.

Additional Tips and Tricks

• Use the property of radicals that $\sqrt{a^2} = a$ to simplify radical expressions.
• Factor the radicand into its prime factors to simplify radical expressions.
• Combine like terms to simplify radical expressions.
• Express the radicand as a product of prime numbers to simplify radical expressions.

Frequently Asked Questions

• What is the final simplified expression of $4 \sqrt{72} + 11 \sqrt{63} - 2 \sqrt{28}$?
  • The final simplified expression is $24 + 29 \times \sqrt{7}$.
• How do I simplify radical expressions?
  • To simplify radical expressions, you need to factor the radicand into its prime factors and use the property of radicals that $\sqrt{a^2} = a$ to simplify the resulting expression.

Final Thoughts

Simplifying radical expressions is a crucial skill in mathematics, and it requires a thorough understanding of the properties of radicals. By factoring the radicands and simplifying the resulting expressions, we can obtain the final simplified expression. In this article, we provided a Q&A section to address any questions or concerns that readers may have. We covered topics such as simplifying radical expressions, factoring the radicand into its prime factors, and combining like terms to simplify radical expressions.