What Is The Value Of The Expression N + 16 \sqrt{N+16} N + 16 ​ When N = 36 N=36 N = 36 ?A. 26 B. 2 13 2 \sqrt{13} 2 13 ​ C. 4 13 4 \sqrt{13} 4 13 ​ D. 4 5 4 \sqrt{5} 4 5 ​ E. 10

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Understanding the Problem

To find the value of the expression $\sqrt{N+16}$ when $N=36$, we need to substitute the value of $N$ into the expression and simplify it. This involves basic algebraic operations and understanding of square roots.

Substituting the Value of $N$

Given that $N=36$, we can substitute this value into the expression $\sqrt{N+16}$ to get $\sqrt{36+16}$. This simplifies to $\sqrt{52}$.

Simplifying the Square Root

To simplify the square root of $52$, we need to find the largest perfect square that divides $52$. In this case, $52$ can be expressed as $4 \times 13$, where $4$ is a perfect square. Therefore, we can rewrite $\sqrt{52}$ as $\sqrt{4 \times 13}$.

Applying the Square Root Property

Using the property of square roots, we can rewrite $\sqrt{4 \times 13}$ as $\sqrt{4} \times \sqrt{13}$. Since $\sqrt{4}$ is equal to $2$, we can simplify this further to $2 \times \sqrt{13}$.

Evaluating the Expression

Therefore, the value of the expression $\sqrt{N+16}$ when $N=36$ is $2 \times \sqrt{13}$.

Comparison with Answer Choices

Comparing this result with the answer choices provided, we can see that the correct answer is $2 \times \sqrt{13}$, which is option B.

Conclusion

In conclusion, to find the value of the expression $\sqrt{N+16}$ when $N=36$, we need to substitute the value of $N$ into the expression and simplify it using basic algebraic operations and understanding of square roots. The correct answer is $2 \times \sqrt{13}$, which is option B.

Final Answer

The final answer is $\boxed{2 \sqrt{13}}$.

Additional Tips and Tricks

• When simplifying square roots, look for the largest perfect square that divides the number inside the square root.
• Use the property of square roots to rewrite the expression as the product of two square roots.
• Simplify the expression by evaluating the square root of the perfect square.

Common Mistakes to Avoid

• Failing to substitute the value of $N$ into the expression.
• Not simplifying the square root properly.
• Not using the property of square roots to rewrite the expression.

Real-World Applications

• Understanding how to simplify square roots is essential in various mathematical and scientific applications, such as physics, engineering, and computer science.
• The ability to simplify square roots is also important in finance, economics, and other fields where mathematical modeling is used.

Practice Problems

• Find the value of the expression $\sqrt{N+25}$ when $N=49$.
• Simplify the square root of $75$.
• Find the value of the expression $\sqrt{N-9}$ when $N=81$.

Solutions to Practice Problems

• The value of the expression $\sqrt{N+25}$ when $N=49$ is $\sqrt{74}$, which can be simplified to $2 \times \sqrt{37}$.
• The square root of $75$ can be simplified to $5 \times \sqrt{3}$.
• The value of the expression $\sqrt{N-9}$ when $N=81$ is $\sqrt{72}$, which can be simplified to $6 \times \sqrt{2}$.

Conclusion

In conclusion, understanding how to simplify square roots is essential in various mathematical and scientific applications. By following the steps outlined in this article, you can simplify square roots and evaluate expressions involving square roots.

Frequently Asked Questions

Q: What is the value of the expression $\sqrt{N+16}$ when $N=36$?

A: The value of the expression $\sqrt{N+16}$ when $N=36$ is $2 \times \sqrt{13}$.

Q: How do I simplify the square root of a number?

A: To simplify the square root of a number, look for the largest perfect square that divides the number inside the square root. Use the property of square roots to rewrite the expression as the product of two square roots. Simplify the expression by evaluating the square root of the perfect square.

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

A: A perfect square is a number that can be expressed as the square of an integer, such as $4$ or $9$. A perfect cube is a number that can be expressed as the cube of an integer, such as $8$ or $27$.

Q: How do I simplify the square root of a fraction?

A: To simplify the square root of a fraction, simplify the fraction first. Then, simplify the square root of the numerator and denominator separately. Finally, simplify the resulting expression.

Q: What is the value of the expression $\sqrt{N+25}$ when $N=49$?

A: The value of the expression $\sqrt{N+25}$ when $N=49$ is $\sqrt{74}$, which can be simplified to $2 \times \sqrt{37}$.

Q: How do I simplify the square root of a decimal number?

A: To simplify the square root of a decimal number, convert the decimal number to a fraction. Then, simplify the square root of the fraction using the steps outlined above.

Q: What is the value of the expression $\sqrt{N-9}$ when $N=81$?

A: The value of the expression $\sqrt{N-9}$ when $N=81$ is $\sqrt{72}$, which can be simplified to $6 \times \sqrt{2}$.

Q: Can I simplify the square root of a negative number?

A: No, you cannot simplify the square root of a negative number. The square root of a negative number is an imaginary number, which is a complex number that cannot be expressed as a real number.

Q: How do I simplify the square root of a binomial expression?

A: To simplify the square root of a binomial expression, use the property of square roots to rewrite the expression as the product of two square roots. Simplify the expression by evaluating the square root of the perfect square.

Q: What is the value of the expression $\sqrt{N+36}$ when $N=64$?

A: The value of the expression $\sqrt{N+36}$ when $N=64$ is $\sqrt{100}$, which can be simplified to $10$.

Q: How do I simplify the square root of a number with multiple factors?

A: To simplify the square root of a number with multiple factors, look for the largest perfect square that divides the number inside the square root. Use the property of square roots to rewrite the expression as the product of two square roots. Simplify the expression by evaluating the square root of the perfect square.

Additional Resources

• For more information on simplifying square roots, see the article "Simplifying Square Roots: A Step-by-Step Guide".
• For practice problems and solutions, see the article "Practice Problems: Simplifying Square Roots".
• For a list of common perfect squares and perfect cubes, see the article "Perfect Squares and Perfect Cubes: A Reference Guide".

Conclusion

In conclusion, simplifying square roots is an essential skill in mathematics and science. By following the steps outlined in this article, you can simplify square roots and evaluate expressions involving square roots. Remember to look for the largest perfect square that divides the number inside the square root, use the property of square roots to rewrite the expression, and simplify the expression by evaluating the square root of the perfect square.