15. Simplify: $(\sqrt{12} - 2\sqrt{3})^2$16. Simplify: $(\sqrt{18} + 2\sqrt{3})^2$17. Simplify: $(\sqrt{5} - \sqrt{6})(\sqrt{5} + \sqrt{9}$\]18. Simplify: $(\sqrt{50} + \sqrt{27})(\sqrt{2} - \sqrt{6}$\]19. Rationalize

Introduction

Radical expressions are an essential part of algebra and mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, focusing on the given problems: 15, 16, 17, 18, and 19. We will delve into the world of square roots, rationalizing denominators, and simplifying complex expressions.

Simplifying Radical Expressions: A Step-by-Step Guide

Simplifying radical expressions involves several steps, including factoring, multiplying, and simplifying. Here's a step-by-step guide to help you simplify radical expressions:

1. Factor the radicand: The radicand is the number inside the square root symbol. Factor the radicand to identify perfect squares.
2. Simplify the square root: Simplify the square root by taking the square root of the perfect squares.
3. Multiply and simplify: Multiply the simplified square roots and simplify the resulting expression.

Problem 15: Simplify $(\sqrt{12} - 2\sqrt{3})^2$

To simplify this expression, we need to follow the steps outlined above.

Step 1: Factor the radicand

The radicand is 12, which can be factored as $4 \times 3$. Since 4 is a perfect square, we can simplify the square root of 12 as $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$.

Step 2: Simplify the square root

Now that we have simplified the square root of 12, we can substitute it back into the original expression: $(2\sqrt{3} - 2\sqrt{3})^2$.

Step 3: Multiply and simplify

The expression inside the parentheses is now a difference of two identical terms, which equals zero. Therefore, the simplified expression is $0^2 = 0$.

Problem 16: Simplify $(\sqrt{18} + 2\sqrt{3})^2$

To simplify this expression, we need to follow the steps outlined above.

Step 1: Factor the radicand

The radicand is 18, which can be factored as $9 \times 2$. Since 9 is a perfect square, we can simplify the square root of 18 as $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$.

Step 2: Simplify the square root

Now that we have simplified the square root of 18, we can substitute it back into the original expression: $(3\sqrt{2} + 2\sqrt{3})^2$.

Step 3: Multiply and simplify

To multiply the two terms inside the parentheses, we need to multiply the coefficients and the square roots separately. The coefficient of the first term is 3, and the coefficient of the second term is 2. The square root of the first term is $\sqrt{2}$, and the square root of the second term is $\sqrt{3}$. Therefore, the product of the two terms is $(3 \times 2)(\sqrt{2} \times \sqrt{3}) = 6\sqrt{6}$.

Problem 17: Simplify $(\sqrt{5} - \sqrt{6})(\sqrt{5} + \sqrt{9})$

To simplify this expression, we need to follow the steps outlined above.

Step 1: Factor the radicand

The radicand is 9, which can be factored as $3 \times 3$. Since 9 is a perfect square, we can simplify the square root of 9 as $\sqrt{9} = 3$.

Step 2: Simplify the square root

Now that we have simplified the square root of 9, we can substitute it back into the original expression: $(\sqrt{5} - \sqrt{6})(\sqrt{5} + 3)$.

Step 3: Multiply and simplify

To multiply the two terms inside the parentheses, we need to multiply the coefficients and the square roots separately. The coefficient of the first term is 1, and the coefficient of the second term is 3. The square root of the first term is $\sqrt{5}$, and the square root of the second term is 3. Therefore, the product of the two terms is $(1 \times 3)(\sqrt{5} \times 3) = 3\sqrt{15}$.

Problem 18: Simplify $(\sqrt{50} + \sqrt{27})(\sqrt{2} - \sqrt{6})$

To simplify this expression, we need to follow the steps outlined above.

Step 1: Factor the radicand

The radicand is 50, which can be factored as $25 \times 2$. Since 25 is a perfect square, we can simplify the square root of 50 as $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$.

The radicand is 27, which can be factored as $9 \times 3$. Since 9 is a perfect square, we can simplify the square root of 27 as $\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$.

Step 2: Simplify the square root

Now that we have simplified the square roots of 50 and 27, we can substitute them back into the original expression: $(5\sqrt{2} + 3\sqrt{3})(\sqrt{2} - \sqrt{6})$.

Step 3: Multiply and simplify

To multiply the two terms inside the parentheses, we need to multiply the coefficients and the square roots separately. The coefficient of the first term is 5, and the coefficient of the second term is 3. The square root of the first term is $\sqrt{2}$, and the square root of the second term is $\sqrt{3}$. Therefore, the product of the two terms is $(5 \times 3)(\sqrt{2} \times \sqrt{3}) = 15\sqrt{6}$.

Problem 19: Rationalize the denominator

To rationalize the denominator, we need to multiply the numerator and denominator by the conjugate of the denominator.

Step 1: Identify the conjugate

The conjugate of the denominator is the same expression with the opposite sign: $\sqrt{2} + \sqrt{6}$.

Step 2: Multiply the numerator and denominator

To rationalize the denominator, we need to multiply the numerator and denominator by the conjugate: $\frac{\sqrt{2} - \sqrt{6}}{\sqrt{2} + \sqrt{6}} \times \frac{\sqrt{2} + \sqrt{6}}{\sqrt{2} + \sqrt{6}}$.

Step 3: Simplify the expression

To simplify the expression, we need to multiply the numerators and denominators separately. The numerator is $(\sqrt{2} - \sqrt{6})(\sqrt{2} + \sqrt{6})$, and the denominator is $(\sqrt{2} + \sqrt{6})(\sqrt{2} + \sqrt{6})$. Therefore, the simplified expression is $\frac{(\sqrt{2} - \sqrt{6})(\sqrt{2} + \sqrt{6})}{(\sqrt{2} + \sqrt{6})(\sqrt{2} + \sqrt{6})}$.

Conclusion

Simplifying radical expressions is a crucial skill to master in algebra and mathematics. By following the steps outlined above, we can simplify complex expressions and rationalize denominators. Remember to factor the radicand, simplify the square root, and multiply and simplify the resulting expression. With practice and patience, you will become proficient in simplifying radical expressions and solving complex problems.

Introduction

Radical expressions are an essential part of algebra and mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, focusing on the most common questions and concerns. We will delve into the world of square roots, rationalizing denominators, and simplifying complex expressions.

Q&A: Simplifying Radical Expressions

Q: What is the difference between a radical and a rational number?

A: A radical is an expression that contains a square root, such as $\sqrt{2}$ or $\sqrt{3}$. A rational number is a number that can be expressed as the ratio of two integers, such as $\frac{1}{2}$ or $\frac{3}{4}$.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to follow these steps:

1. Factor the radicand: The radicand is the number inside the square root symbol. Factor the radicand to identify perfect squares.
2. Simplify the square root: Simplify the square root by taking the square root of the perfect squares.
3. Multiply and simplify: Multiply the simplified square roots and simplify the resulting expression.

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 $4 = 2^2$ or $9 = 3^2$. A perfect cube is a number that can be expressed as the cube of an integer, such as $8 = 2^3$ or $27 = 3^3$.

Q: How do I rationalize a denominator?

A: To rationalize a denominator, you need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of the denominator is the same expression with the opposite sign.

Q: What is the difference between a rationalizing denominator and a rationalizing numerator?

A: A rationalizing denominator is the process of multiplying the numerator and denominator by the conjugate of the denominator to eliminate the square root in the denominator. A rationalizing numerator is the process of multiplying the numerator and denominator by the conjugate of the numerator to eliminate the square root in the numerator.

Q: How do I simplify a complex radical expression?

A: To simplify a complex radical expression, you need to follow these steps:

1. Simplify each radical expression separately: Simplify each radical expression separately using the steps outlined above.
2. Multiply and simplify: Multiply the simplified radical expressions and simplify the resulting expression.

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

A: A radical expression is an expression that contains a square root, such as $\sqrt{2}$ or $\sqrt{3}$. A rational expression is an expression that contains a fraction, such as $\frac{1}{2}$ or $\frac{3}{4}$.

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

A: To simplify a radical expression with a variable, you need to follow these steps:

1. Simplify the radical expression: Simplify the radical expression using the steps outlined above.
2. Substitute the variable: Substitute the variable into the simplified radical expression.

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

A: A radical expression is an expression that contains a square root, such as $\sqrt{2}$ or $\sqrt{3}$. An exponential expression is an expression that contains a power, such as $2^3$ or $3^4$.

Conclusion

Simplifying radical expressions is a crucial skill to master in algebra and mathematics. By following the steps outlined above and answering the most common questions and concerns, you will become proficient in simplifying radical expressions and solving complex problems. Remember to factor the radicand, simplify the square root, and multiply and simplify the resulting expression. With practice and patience, you will become a master of simplifying radical expressions.