Simplify The Expression: $\[ 5 \sqrt{7} - 4 \sqrt{2} + 3 \sqrt{7} \\]

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Introduction

When dealing with algebraic expressions that involve radicals, it's essential to understand how to simplify them. In this article, we will focus on simplifying an expression that contains like terms with radicals. We will use the given expression ${ 5 \sqrt{7} - 4 \sqrt{2} + 3 \sqrt{7} }$ as an example and demonstrate how to combine like terms with radicals.

Understanding Like Terms with Radicals

Like terms are terms that have the same variable raised to the same power. In the case of radicals, like terms are terms that have the same radical expression. For example, 232 \sqrt{3} and 333 \sqrt{3} are like terms because they both have the same radical expression 3\sqrt{3}.

Simplifying the Expression

To simplify the given expression, we need to combine the like terms with radicals. The expression is ${ 5 \sqrt{7} - 4 \sqrt{2} + 3 \sqrt{7} }$. We can see that there are two terms with the radical 7\sqrt{7}, which are 575 \sqrt{7} and 373 \sqrt{7}. These two terms are like terms because they both have the same radical expression 7\sqrt{7}.

Combining Like Terms with Radicals

To combine like terms with radicals, we need to add or subtract the coefficients of the like terms. In this case, we need to add the coefficients of the two terms with the radical 7\sqrt{7}. The coefficients are 5 and 3, so we add them together to get 5+3=85 + 3 = 8. Therefore, the simplified expression for the two terms with the radical 7\sqrt{7} is 878 \sqrt{7}.

Simplifying the Expression Further

Now that we have simplified the two terms with the radical 7\sqrt{7}, we can simplify the entire expression. The expression is now ${ 8 \sqrt{7} - 4 \sqrt{2} }$. We can see that there are no other like terms with radicals in the expression, so we are done.

Conclusion

In this article, we simplified an expression that contained like terms with radicals. We used the given expression ${ 5 \sqrt{7} - 4 \sqrt{2} + 3 \sqrt{7} }$ as an example and demonstrated how to combine like terms with radicals. We showed that by adding or subtracting the coefficients of the like terms, we can simplify the expression and obtain a simpler form.

Tips and Tricks

  • When dealing with like terms with radicals, make sure to identify the like terms and combine them.
  • Use the distributive property to simplify expressions with radicals.
  • Be careful when adding or subtracting coefficients of like terms with radicals.

Example Problems

  • Simplify the expression ${ 2 \sqrt{3} + 4 \sqrt{3} - 3 \sqrt{3} }$.
  • Simplify the expression ${ 5 \sqrt{2} - 2 \sqrt{2} + 3 \sqrt{2} }$.

Solutions

  • The simplified expression for the given expression is ${ 3 \sqrt{3} }$.
  • The simplified expression for the given expression is ${ 6 \sqrt{2} }$.

Final Thoughts

Simplifying expressions with like terms with radicals is an essential skill in algebra. By understanding how to combine like terms with radicals, we can simplify complex expressions and obtain a simpler form. In this article, we demonstrated how to simplify an expression that contained like terms with radicals and provided tips and tricks for simplifying expressions with radicals.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak

Further Reading

  • "Simplifying Expressions with Radicals" by Math Open Reference
  • "Like Terms with Radicals" by Purplemath

Related Articles

  • "Simplifying Expressions with Exponents"
  • "Simplifying Expressions with Fractions"

Keywords

  • Simplifying expressions with radicals
  • Like terms with radicals
  • Combining like terms with radicals
  • Algebra
  • Radicals
  • Expressions with radicals
  • Simplifying expressions with exponents
  • Simplifying expressions with fractions

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Frequently Asked Questions

Q: What are like terms with radicals?

A: Like terms with radicals are terms that have the same radical expression. For example, 232 \sqrt{3} and 333 \sqrt{3} are like terms because they both have the same radical expression 3\sqrt{3}.

Q: How do I combine like terms with radicals?

A: To combine like terms with radicals, you need to add or subtract the coefficients of the like terms. For example, if you have the expression 23+332 \sqrt{3} + 3 \sqrt{3}, you can combine the like terms by adding the coefficients: 2+3=52 + 3 = 5. Therefore, the simplified expression is 535 \sqrt{3}.

Q: What is the distributive property in relation to radicals?

A: The distributive property is a rule that allows you to multiply a single term by multiple terms. In the context of radicals, the distributive property can be used to simplify expressions with radicals. For example, if you have the expression 23(4+5)2 \sqrt{3} (4 + 5), you can use the distributive property to simplify the expression: 23(4+5)=23(9)=1832 \sqrt{3} (4 + 5) = 2 \sqrt{3} (9) = 18 \sqrt{3}.

Q: Can I simplify an expression with radicals that has a negative coefficient?

A: Yes, you can simplify an expression with radicals that has a negative coefficient. For example, if you have the expression −23-2 \sqrt{3}, you can simplify it by keeping the negative sign and simplifying the radical expression: −23-2 \sqrt{3}.

Q: How do I simplify an expression with radicals that has multiple terms with different radicals?

A: To simplify an expression with radicals that has multiple terms with different radicals, you need to identify the like terms and combine them. For example, if you have the expression 23+32−432 \sqrt{3} + 3 \sqrt{2} - 4 \sqrt{3}, you can identify the like terms and combine them: 23−43=−232 \sqrt{3} - 4 \sqrt{3} = -2 \sqrt{3} and 323 \sqrt{2} is a like term with no other like terms, so it remains the same. Therefore, the simplified expression is −23+32-2 \sqrt{3} + 3 \sqrt{2}.

Q: Can I simplify an expression with radicals that has a variable in the radical expression?

A: Yes, you can simplify an expression with radicals that has a variable in the radical expression. For example, if you have the expression 2x+3x2 \sqrt{x} + 3 \sqrt{x}, you can simplify it by combining the like terms: 2x+3x=5x2 \sqrt{x} + 3 \sqrt{x} = 5 \sqrt{x}.

Additional Tips and Tricks

  • When dealing with like terms with radicals, make sure to identify the like terms and combine them.
  • Use the distributive property to simplify expressions with radicals.
  • Be careful when adding or subtracting coefficients of like terms with radicals.
  • Simplify expressions with radicals that have variables in the radical expression by combining like terms.

Example Problems

  • Simplify the expression ${ 2 \sqrt{3} + 4 \sqrt{3} - 3 \sqrt{3} }$.
  • Simplify the expression ${ 5 \sqrt{2} - 2 \sqrt{2} + 3 \sqrt{2} }$.
  • Simplify the expression ${ 2 \sqrt{x} + 3 \sqrt{x} - 4 \sqrt{x} }$.

Solutions

  • The simplified expression for the given expression is ${ 2 \sqrt{3} }$.
  • The simplified expression for the given expression is ${ 6 \sqrt{2} }$.
  • The simplified expression for the given expression is ${ - \sqrt{x} }$.

Final Thoughts

Simplifying expressions with radicals is an essential skill in algebra. By understanding how to combine like terms with radicals, we can simplify complex expressions and obtain a simpler form. In this article, we provided answers to frequently asked questions about simplifying expressions with radicals and provided additional tips and tricks for simplifying expressions with radicals.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak

Further Reading

  • "Simplifying Expressions with Radicals" by Math Open Reference
  • "Like Terms with Radicals" by Purplemath

Related Articles

  • "Simplifying Expressions with Exponents"
  • "Simplifying Expressions with Fractions"

Keywords

  • Simplifying expressions with radicals
  • Like terms with radicals
  • Combining like terms with radicals
  • Algebra
  • Radicals
  • Expressions with radicals
  • Simplifying expressions with exponents
  • Simplifying expressions with fractions