Simplify The Expression:${ 5 \sqrt{5} - 3 \sqrt{28} - (8 \sqrt{20} + 3 \sqrt{7}) }$

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Introduction

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Simplifying radical expressions is an essential skill in mathematics, particularly in algebra and geometry. It involves combining like terms and simplifying the expression to its simplest form. In this article, we will simplify the given expression: $5 \sqrt{5} - 3 \sqrt{28} - (8 \sqrt{20} + 3 \sqrt{7})$. We will break down the steps involved in simplifying the expression and provide a clear explanation of each step.

Understanding the Expression

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The given expression is a combination of several terms, each containing a square root. To simplify the expression, we need to understand the properties of square roots and how to combine like terms.

Properties of Square Roots

Radical Notation

The square root of a number is denoted by the symbol $\sqrt{}$. For example, $\sqrt{16}$ represents the number that, when multiplied by itself, gives 16.

Simplifying Square Roots

We can simplify square roots by finding the largest perfect square that divides the number inside the square root. For example, $\sqrt{16}$ can be simplified as $\sqrt{16} = \sqrt{4 \times 4} = 4$.

Combining Like Terms

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Like terms are terms that have the same variable and exponent. In the given expression, we have several like terms that we can combine to simplify the expression.

Step 1: Simplify the Square Roots

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The first step in simplifying the expression is to simplify the square roots. We can start by simplifying $\sqrt{28}$ and $\sqrt{20}$.

Simplifying $\sqrt{28}$

$\sqrt{28}$ can be simplified as $\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$.

Simplifying $\sqrt{20}$

$\sqrt{20}$ can be simplified as $\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$.

Step 2: Substitute the Simplified Square Roots

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Now that we have simplified the square roots, we can substitute the simplified expressions back into the original expression.

Substituting $\sqrt{28}$ and $\sqrt{20}$

The original expression becomes: $5 \sqrt{5} - 3(2\sqrt{7}) - (8(2\sqrt{5}) + 3 \sqrt{7})$.

Step 3: Distribute the Negative Sign

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The next step is to distribute the negative sign to the terms inside the parentheses.

Distributing the Negative Sign

The expression becomes: $5 \sqrt{5} - 6\sqrt{7} - 16\sqrt{5} - 3 \sqrt{7}$.

Step 4: Combine Like Terms

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Now that we have distributed the negative sign, we can combine like terms.

Combining Like Terms

The expression becomes: $5 \sqrt{5} - 16\sqrt{5} - 6\sqrt{7} - 3 \sqrt{7}$.

Step 5: Simplify the Expression

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The final step is to simplify the expression by combining like terms.

Simplifying the Expression

The expression becomes: $-11\sqrt{5} - 9\sqrt{7}$.

Conclusion

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Simplifying radical expressions involves combining like terms and simplifying the expression to its simplest form. In this article, we simplified the given expression: $5 \sqrt{5} - 3 \sqrt{28} - (8 \sqrt{20} + 3 \sqrt{7})$. We broke down the steps involved in simplifying the expression and provided a clear explanation of each step. By following these steps, you can simplify any radical expression and arrive at the simplest form.

Frequently Asked Questions

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Q: What is the difference between a radical and a square root?

A: A radical is a mathematical expression that involves a square root. For example, $\sqrt{16}$ is a radical expression.

Q: How do I simplify a square root?

A: To simplify a square root, find the largest perfect square that divides the number inside the square root.

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable and exponent. Unlike terms are terms that have different variables or exponents.

Final Thoughts

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Simplifying radical expressions is an essential skill in mathematics, particularly in algebra and geometry. By following the steps outlined in this article, you can simplify any radical expression and arrive at the simplest form. Remember to simplify the square roots, substitute the simplified expressions back into the original expression, distribute the negative sign, combine like terms, and simplify the expression. With practice and patience, you can become proficient in simplifying radical expressions and tackle even the most complex mathematical problems.

Additional Resources

• Mathway: Simplifying Radical Expressions
• Khan Academy: Simplifying Radical Expressions
• Purplemath: Simplifying Radical Expressions

Key Takeaways

• Simplifying radical expressions involves combining like terms and simplifying the expression to its simplest form.
• To simplify a square root, find the largest perfect square that divides the number inside the square root.
• Like terms are terms that have the same variable and exponent.
• Unlike terms are terms that have different variables or exponents.

Conclusion

Simplifying radical expressions is an essential skill in mathematics, particularly in algebra and geometry. By following the steps outlined in this article, you can simplify any radical expression and arrive at the simplest form. Remember to simplify the square roots, substitute the simplified expressions back into the original expression, distribute the negative sign, combine like terms, and simplify the expression. With practice and patience, you can become proficient in simplifying radical expressions and tackle even the most complex mathematical problems.

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Introduction

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Simplifying radical expressions is an essential skill in mathematics, particularly in algebra and geometry. In our previous article, we provided a step-by-step guide to simplifying the expression: $5 \sqrt{5} - 3 \sqrt{28} - (8 \sqrt{20} + 3 \sqrt{7})$. In this article, we will answer some frequently asked questions about simplifying radical expressions.

Q&A

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Q: What is the difference between a radical and a square root?

A: A radical is a mathematical expression that involves a square root. For example, $\sqrt{16}$ is a radical expression. A square root is a number that, when multiplied by itself, gives the original number. For example, $\sqrt{16} = 4$ because $4 \times 4 = 16$.

Q: How do I simplify a square root?

A: To simplify a square root, find the largest perfect square that divides the number inside the square root. For example, $\sqrt{16}$ can be simplified as $\sqrt{16} = \sqrt{4 \times 4} = 4$.

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable and exponent. Unlike terms are terms that have different variables or exponents. For example, $2x^2$ and $3x^2$ are like terms because they have the same variable and exponent, while $2x^2$ and $3y^2$ are unlike terms because they have different variables.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the like terms. For example, $2x^2 + 3x^2$ can be combined as $5x^2$.

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

A: A rational expression is an expression that can be written as a fraction of two polynomials. An irrational expression is an expression that cannot be written as a fraction of two polynomials. For example, $\frac{1}{2}$ is a rational expression, while $\sqrt{2}$ is an irrational expression.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, factor the numerator and denominator, and then cancel out any common factors. For example, $\frac{2x}{2x}$ can be simplified as $1$.

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

A: A radical expression is an expression that involves a square root. A rational expression is an expression that can be written as a fraction of two polynomials. For example, $\sqrt{2}$ is a radical expression, while $\frac{1}{2}$ is a rational expression.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, find the largest perfect square that divides the number inside the square root, and then simplify the expression. For example, $\sqrt{16}$ can be simplified as $\sqrt{16} = \sqrt{4 \times 4} = 4$.

Q: What is the difference between a conjugate and a reciprocal?

A: A conjugate is an expression that is the same as the original expression, but with the opposite sign. A reciprocal is an expression that is the same as the original expression, but with the numerator and denominator swapped. For example, the conjugate of $2x + 3$ is $2x - 3$, while the reciprocal of $2x + 3$ is $\frac{1}{2x + 3}$.

Q: How do I simplify a conjugate?

A: To simplify a conjugate, add or subtract the expressions. For example, $(2x + 3) + (2x - 3)$ can be simplified as $4x$.

Q: What is the difference between a rational exponent and an irrational exponent?

A: A rational exponent is an exponent that can be written as a fraction of two integers. An irrational exponent is an exponent that cannot be written as a fraction of two integers. For example, $x^{\frac{1}{2}}$ is a rational exponent, while $x^{\sqrt{2}}$ is an irrational exponent.

Q: How do I simplify a rational exponent?

A: To simplify a rational exponent, raise the base to the power of the numerator, and then take the root of the denominator. For example, $x^{\frac{1}{2}}$ can be simplified as $\sqrt{x}$.

Conclusion

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Simplifying radical expressions is an essential skill in mathematics, particularly in algebra and geometry. By following the steps outlined in this article, you can simplify any radical expression and arrive at the simplest form. Remember to simplify the square roots, substitute the simplified expressions back into the original expression, distribute the negative sign, combine like terms, and simplify the expression. With practice and patience, you can become proficient in simplifying radical expressions and tackle even the most complex mathematical problems.

Additional Resources

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• Mathway: Simplifying Radical Expressions
• Khan Academy: Simplifying Radical Expressions
• Purplemath: Simplifying Radical Expressions

Key Takeaways

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• Simplifying radical expressions involves combining like terms and simplifying the expression to its simplest form.
• To simplify a square root, find the largest perfect square that divides the number inside the square root.
• Like terms are terms that have the same variable and exponent.
• Unlike terms are terms that have different variables or exponents.
• Rational expressions are expressions that can be written as a fraction of two polynomials.
• Irrational expressions are expressions that cannot be written as a fraction of two polynomials.
• Conjugates are expressions that are the same as the original expression, but with the opposite sign.
• Reciprocals are expressions that are the same as the original expression, but with the numerator and denominator swapped.