Which Expression Is Equivalent To 300 + 5 3 \sqrt{300} + 5 \sqrt{3} 300 ​ + 5 3 ​ ?A. 35 13 35 \sqrt{13} 35 13 ​ B. 15 3 15 \sqrt{3} 15 3 ​ C. 3 10 + 5 3 3 \sqrt{10} + 5 \sqrt{3} 3 10 ​ + 5 3 ​ D. None Of These E. 8 13 8 \sqrt{13} 8 13 ​

by ADMIN 239 views

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression 300+53\sqrt{300} + 5 \sqrt{3}. We will examine each option and determine which one is equivalent to the given expression.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or other root. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

Breaking Down the Given Expression

The given expression is 300+53\sqrt{300} + 5 \sqrt{3}. To simplify this expression, we need to break it down into its prime factors. We can start by factoring 300 into its prime factors:

300 = 2 × 2 × 3 × 5 × 5

We can rewrite the expression as:

22×3×52+53\sqrt{2^2 × 3 × 5^2} + 5 \sqrt{3}

Simplifying the Square Root

Now that we have factored the expression, we can simplify the square root. We can take the square root of the perfect squares, which are the numbers that have an even exponent:

22×3×52=(22)×(52)×3\sqrt{2^2 × 3 × 5^2} = \sqrt{(2^2) × (5^2)} × \sqrt{3}

=2×5×3= 2 × 5 × \sqrt{3}

=103= 10 \sqrt{3}

Combining Like Terms

Now that we have simplified the square root, we can combine like terms. We have 10310 \sqrt{3} and 535 \sqrt{3}, which are like terms because they have the same variable and exponent. We can combine them by adding their coefficients:

103+53=15310 \sqrt{3} + 5 \sqrt{3} = 15 \sqrt{3}

Evaluating the Options

Now that we have simplified the expression, we can evaluate the options. We have:

A. 351335 \sqrt{13} B. 15315 \sqrt{3} C. 310+533 \sqrt{10} + 5 \sqrt{3} D. None of these E. 8138 \sqrt{13}

Option A

Option A is 351335 \sqrt{13}. We can see that this option is not equivalent to the given expression, because it has a different variable and exponent.

Option B

Option B is 15315 \sqrt{3}. We can see that this option is equivalent to the given expression, because it has the same variable and exponent.

Option C

Option C is 310+533 \sqrt{10} + 5 \sqrt{3}. We can see that this option is not equivalent to the given expression, because it has a different variable and exponent.

Option D

Option D is None of these. We can see that this option is incorrect, because option B is equivalent to the given expression.

Option E

Option E is 8138 \sqrt{13}. We can see that this option is not equivalent to the given expression, because it has a different variable and exponent.

Conclusion

In conclusion, the correct answer is option B, 15315 \sqrt{3}. This option is equivalent to the given expression, because it has the same variable and exponent. We can see that simplifying radical expressions is a crucial skill to master, and it requires a deep understanding of the underlying mathematics.

Final Answer

The final answer is B\boxed{B}.

Additional Resources

For more information on simplifying radical expressions, please see the following resources:

References

Introduction

In our previous article, we explored the process of simplifying radical expressions, with a focus on the given expression 300+53\sqrt{300} + 5 \sqrt{3}. We determined that the correct answer is option B, 15315 \sqrt{3}. In this article, we will provide a Q&A guide to help you better understand the concept of simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a square root or other root. The square root of a number is a value that, when multiplied by itself, gives the original number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to break it down into its prime factors. You can then take the square root of the perfect squares, which are the numbers that have an even exponent.

Q: What is a perfect square?

A: A perfect square is a number that has an even exponent. For example, 4 is a perfect square because it can be expressed as 2^2.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms. For example, if you have 2x and 3x, you can combine them by adding their coefficients: 2x + 3x = 5x.

Q: What is the difference between a radical expression and an exponential expression?

A: A radical expression is a mathematical expression that contains a square root or other root, while an exponential expression is a mathematical expression that contains a power or exponent. For example, 2^3 is an exponential expression, while √2 is a radical expression.

Q: Can I simplify a radical expression with a variable?

A: Yes, you can simplify a radical expression with a variable. For example, if you have √(x^2), you can simplify it by taking the square root of the perfect square: √(x^2) = x.

Q: How do I simplify a radical expression with a coefficient?

A: To simplify a radical expression with a coefficient, you need to multiply the coefficient by the square root of the number. For example, if you have 2√3, you can simplify it by multiplying the coefficient by the square root of the number: 2√3 = 2√3.

Q: Can I simplify a radical expression with a negative number?

A: Yes, you can simplify a radical expression with a negative number. For example, if you have -√3, you can simplify it by multiplying the negative sign by the square root of the number: -√3 = -√3.

Q: How do I simplify a radical expression with a fraction?

A: To simplify a radical expression with a fraction, you need to multiply the numerator and denominator by the square root of the number. For example, if you have √(1/4), you can simplify it by multiplying the numerator and denominator by the square root of the number: √(1/4) = √(1/4) = 1/2.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill to master, and it requires a deep understanding of the underlying mathematics. By following the steps outlined in this article, you can simplify radical expressions with ease.

Final Tips

  • Always break down the radical expression into its prime factors.
  • Take the square root of the perfect squares.
  • Combine like terms by adding or subtracting the coefficients.
  • Simplify radical expressions with variables, coefficients, and negative numbers.
  • Simplify radical expressions with fractions by multiplying the numerator and denominator by the square root of the number.

Additional Resources

For more information on simplifying radical expressions, please see the following resources:

References