Multiply the coefficient by the simplified radical

Oct 13, 2025 by windailys.com 118 views

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the process of simplifying radical expressions, focusing on the given examples: $4 \sqrt$ , $7 \sqrt[3]{x^2 - 2 \sqrt[3]$ , $\sqrt[3]{54 + \sqrt[3]$ , $\sqrt[4]{32 + \sqrt[4]$ , $(\sqrt{13 + 6)^2$ , and $(3 - 4\sqrt{2})^2$ .

Simplifying Radical Expressions: A Step-by-Step Approach

Simplifying radical expressions involves combining like terms, factoring, and using the properties of radicals. Here are the steps to simplify radical expressions:

Identify like terms: Look for terms that have the same radical expression.
Combine like terms: Add or subtract the coefficients of like terms.
Factor the radicand: Factor the expression inside the radical sign.
Simplify the radical: Use the properties of radicals to simplify the expression.

Example a: Simplifying $4 \sqrt{3} + 4 \sqrt[3]{3}$

To simplify the expression $4 \sqrt{3} + 4 \sqrt[3]{3}$ , we need to combine like terms and factor the radicand.

$4 \sqrt{3} + 4 \sqrt[3]{3}$

We can rewrite the expression as:

$4(\sqrt{3} + \sqrt[3]{3})$

Now, we can simplify the expression by factoring the radicand:

$4(\sqrt{3} + \sqrt[3]{3}) = 4\sqrt{3} + 4\sqrt[3]{3}$

However, we can simplify the expression further by recognizing that $\sqrt{3}$ and $\sqrt[3]{3}$ are both cube roots of 3. Therefore, we can rewrite the expression as:

$4\sqrt{3} + 4\sqrt[3]{3} = 4(\sqrt{3} + \sqrt[3]{3}) = 4\sqrt[3]{3^2} = 4\sqrt[3]{9}$

Example b: Simplifying $7 \sqrt[3]{x^2} - 2 \sqrt[3]{x^2}$

To simplify the expression $7 \sqrt[3]{x^2} - 2 \sqrt[3]{x^2}$ , we need to combine like terms.

$7 \sqrt[3]{x^2} - 2 \sqrt[3]{x^2}$

We can rewrite the expression as:

$(7 - 2)\sqrt[3]{x^2} = 5\sqrt[3]{x^2}$

Example c: Simplifying $\sqrt[3]{54} + \sqrt[3]{16}$

To simplify the expression $\sqrt[3]{54} + \sqrt[3]{16}$ , we need to factor the radicand.

$\sqrt[3]{54} + \sqrt[3]{16}$

We can rewrite the expression as:

$\sqrt[3]{54} + \sqrt[3]{16} = \sqrt[3]{2\cdot 3^3} + \sqrt[3]{2^3} = 3\sqrt[3]{2} + 2\sqrt[3]{2} = 5\sqrt[3]{2}$

Example d: Simplifying $\sqrt[4]{32} + \sqrt[4]{48}$

To simplify the expression $\sqrt[4]{32} + \sqrt[4]{48}$ , we need to factor the radicand.

$\sqrt[4]{32} + \sqrt[4]{48}$

We can rewrite the expression as:

$\sqrt[4]{32} + \sqrt[4]{48} = \sqrt[4]{2^5} + \sqrt[4]{2^4 \cdot 3} = 2\sqrt[4]{2} + \sqrt[4]{2^4 \cdot 3} = 2\sqrt[4]{2} + 2\sqrt[4]{3}$

Example e: Simplifying $(\sqrt{13} + 6)^2$

To simplify the expression $(\sqrt{13} + 6)^2$ , we need to expand the square.

$(\sqrt{13} + 6)^2$

We can rewrite the expression as:

$(\sqrt{13} + 6)^2 = (\sqrt{13})^2 + 2(\sqrt{13})(6) + 6^2 = 13 + 12\sqrt{13} + 36$

Example f: Simplifying $(3 - 4\sqrt{2})^2$

To simplify the expression $(3 - 4\sqrt{2})^2$ , we need to expand the square.

$(3 - 4\sqrt{2})^2$

We can rewrite the expression as:

$(3 - 4\sqrt{2})^2 = (3)^2 - 2(3)(4\sqrt{2}) + (4\sqrt{2})^2 = 9 - 24\sqrt{2} + 32$

Conclusion

Simplifying radical expressions is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can simplify radical expressions and solve problems with ease. Remember to identify like terms, combine like terms, factor the radicand, and simplify the radical. With practice and patience, you can master the art of simplifying radical expressions.

References

[1] "Simplifying Radical Expressions" by Math Open Reference
[2] "Radical Expressions" by Khan Academy
[3] "Simplifying Radical Expressions" by Purplemath

Q: What is a radical expression?

A: A radical expression is an expression that contains a radical sign, which is denoted by the symbol $\sqrt{}$ . Radical expressions can be simplified by combining like terms, factoring the radicand, and using the properties of radicals.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, follow these steps:

Identify like terms: Look for terms that have the same radical expression.
Combine like terms: Add or subtract the coefficients of like terms.
Factor the radicand: Factor the expression inside the radical sign.
Simplify the radical: Use the properties of radicals to simplify the expression.

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

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., $a/b$ , where $a$ and $b$ are integers and $b$ is non-zero. An irrational number is a number that cannot be expressed as the ratio of two integers.

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

A: To simplify a radical expression with a rational coefficient, follow these steps:

Factor the radicand: Factor the expression inside the radical sign.
Simplify the radical: Use the properties of radicals to simplify the expression.
Multiply the coefficient by the simplified radical: Multiply the rational coefficient by the simplified radical.

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

A: To simplify a radical expression with an irrational coefficient, follow these steps:

Factor the radicand: Factor the expression inside the radical sign.
Simplify the radical: Use the properties of radicals to simplify the expression.
Leave the irrational coefficient as is: The irrational coefficient cannot be simplified further.

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, i.e., $n^2$ , where $n$ is an integer. A perfect cube is a number that can be expressed as the cube of an integer, i.e., $n^3$ , where $n$ is an integer.

Q: How do I simplify a radical expression with a perfect square or perfect cube?

A: To simplify a radical expression with a perfect square or perfect cube, follow these steps:

Factor the radicand: Factor the expression inside the radical sign.
Simplify the radical: Use the properties of radicals to simplify the expression.
Remove the perfect square or perfect cube: Remove the perfect square or perfect cube from the radicand.

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

A: A radical expression is an expression that contains a radical sign, which is denoted by the symbol $\sqrt{}$ . An exponential expression is an expression that contains an exponent, which is denoted by the symbol $a^b$ , where $a$ is the base and $b$ is the exponent.

Q: How do I simplify a radical expression with an exponential expression?

A: To simplify a radical expression with an exponential expression, follow these steps:

Simplify the radical expression: Simplify the radical expression using the properties of radicals.
Simplify the exponential expression: Simplify the exponential expression using the properties of exponents.
Combine the simplified expressions: Combine the simplified radical and exponential expressions.

Conclusion

References

[1] "Simplifying Radical Expressions" by Math Open Reference
[2] "Radical Expressions" by Khan Academy
[3] "Simplifying Radical Expressions" by Purplemath