Express In Simplest Radical Form: 25 X 13 \sqrt{25 X^{13}} 25 X 13 ​ Answer:A. 5 X 6 X 5 X^6 \sqrt{x} 5 X 6 X ​ B. X 6 25 X X^6 \sqrt{25 X} X 6 25 X ​ C. 5 X 13 5 \sqrt{x^{13}} 5 X 13 ​

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Introduction

Radicals are an essential part of mathematics, and expressing them in their simplest form is crucial for solving equations and inequalities. In this article, we will focus on expressing the expression 25x13\sqrt{25 x^{13}} in its simplest radical form. We will explore the properties of radicals, provide step-by-step solutions, and offer tips and tricks for simplifying radicals.

Understanding Radicals

A radical is a mathematical expression that represents a quantity that can be expressed as a power of a number. The most common type of radical is the square root, which is denoted by the symbol \sqrt{ }. The square root of a number is a value that, when multiplied by itself, gives the original number.

Properties of Radicals

There are several properties of radicals that we need to understand in order to simplify them. These properties include:

  • Product Property: ab=a⋅b\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}
  • Quotient Property: ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}
  • Power Property: ab=ab2\sqrt{a^b} = a^{\frac{b}{2}}

Simplifying Radicals

Now that we have a good understanding of the properties of radicals, let's apply them to simplify the expression 25x13\sqrt{25 x^{13}}.

Step 1: Factor the Number Under the Radical

The first step in simplifying the expression is to factor the number under the radical. In this case, we can factor 25 as 525^2.

\sqrt{25 x^{13}} = \sqrt{5^2 x^{13}}

Step 2: Apply the Product Property

Next, we can apply the product property to separate the factors under the radical.

\sqrt{5^2 x^{13}} = \sqrt{5^2} \cdot \sqrt{x^{13}}

Step 3: Simplify the Square Root of the Number

Now, we can simplify the square root of the number by taking the square root of the factor.

\sqrt{5^2} = 5

Step 4: Apply the Power Property

Next, we can apply the power property to simplify the square root of the variable.

\sqrt{x^{13}} = x^{\frac{13}{2}}

Step 5: Simplify the Expression

Finally, we can simplify the expression by combining the simplified factors.

5 \cdot x^{\frac{13}{2}} = 5 x^6 \sqrt{x}

Conclusion

In this article, we have learned how to express the expression 25x13\sqrt{25 x^{13}} in its simplest radical form. We have applied the properties of radicals, including the product property, quotient property, and power property, to simplify the expression. We have also provided step-by-step solutions and offered tips and tricks for simplifying radicals.

Tips and Tricks

Here are some tips and tricks for simplifying radicals:

  • Factor the number under the radical: Factor the number under the radical to simplify the expression.
  • Apply the product property: Apply the product property to separate the factors under the radical.
  • Simplify the square root of the number: Simplify the square root of the number by taking the square root of the factor.
  • Apply the power property: Apply the power property to simplify the square root of the variable.
  • Combine the simplified factors: Combine the simplified factors to simplify the expression.

Common Mistakes

Here are some common mistakes to avoid when simplifying radicals:

  • Not factoring the number under the radical: Failing to factor the number under the radical can make it difficult to simplify the expression.
  • Not applying the product property: Failing to apply the product property can make it difficult to separate the factors under the radical.
  • Not simplifying the square root of the number: Failing to simplify the square root of the number can make it difficult to simplify the expression.
  • Not applying the power property: Failing to apply the power property can make it difficult to simplify the square root of the variable.
  • Not combining the simplified factors: Failing to combine the simplified factors can make it difficult to simplify the expression.

Final Answer

Q&A: Simplifying Radicals

In this article, we will provide answers to some of the most frequently asked questions about simplifying radicals.

Q: What is the simplest radical form of 16x8\sqrt{16 x^8}?

A: To simplify the expression, we can factor the number under the radical as 424^2. Then, we can apply the product property to separate the factors under the radical.

\sqrt{16 x^8} = \sqrt{4^2 x^8} = \sqrt{4^2} \cdot \sqrt{x^8} = 4 \cdot x^4 = 4 x^4

Q: What is the simplest radical form of 9x7\sqrt{9 x^7}?

A: To simplify the expression, we can factor the number under the radical as 323^2. Then, we can apply the product property to separate the factors under the radical.

\sqrt{9 x^7} = \sqrt{3^2 x^7} = \sqrt{3^2} \cdot \sqrt{x^7} = 3 \cdot x^{\frac{7}{2}} = 3 x^{\frac{7}{2}}

Q: What is the simplest radical form of 36x9\sqrt{36 x^9}?

A: To simplify the expression, we can factor the number under the radical as 626^2. Then, we can apply the product property to separate the factors under the radical.

\sqrt{36 x^9} = \sqrt{6^2 x^9} = \sqrt{6^2} \cdot \sqrt{x^9} = 6 \cdot x^{\frac{9}{2}} = 6 x^{\frac{9}{2}}

Q: What is the simplest radical form of 25x5\sqrt{25 x^5}?

A: To simplify the expression, we can factor the number under the radical as 525^2. Then, we can apply the product property to separate the factors under the radical.

\sqrt{25 x^5} = \sqrt{5^2 x^5} = \sqrt{5^2} \cdot \sqrt{x^5} = 5 \cdot x^{\frac{5}{2}} = 5 x^{\frac{5}{2}}

Q: What is the simplest radical form of 49x3\sqrt{49 x^3}?

A: To simplify the expression, we can factor the number under the radical as 727^2. Then, we can apply the product property to separate the factors under the radical.

\sqrt{49 x^3} = \sqrt{7^2 x^3} = \sqrt{7^2} \cdot \sqrt{x^3} = 7 \cdot x^{\frac{3}{2}} = 7 x^{\frac{3}{2}}

Q: What is the simplest radical form of 81x6\sqrt{81 x^6}?

A: To simplify the expression, we can factor the number under the radical as 929^2. Then, we can apply the product property to separate the factors under the radical.

\sqrt{81 x^6} = \sqrt{9^2 x^6} = \sqrt{9^2} \cdot \sqrt{x^6} = 9 \cdot x^3 = 9 x^3

Conclusion

In this article, we have provided answers to some of the most frequently asked questions about simplifying radicals. We have applied the properties of radicals, including the product property, quotient property, and power property, to simplify the expressions. We have also provided step-by-step solutions and offered tips and tricks for simplifying radicals.

Tips and Tricks

Here are some tips and tricks for simplifying radicals:

  • Factor the number under the radical: Factor the number under the radical to simplify the expression.
  • Apply the product property: Apply the product property to separate the factors under the radical.
  • Simplify the square root of the number: Simplify the square root of the number by taking the square root of the factor.
  • Apply the power property: Apply the power property to simplify the square root of the variable.
  • Combine the simplified factors: Combine the simplified factors to simplify the expression.

Common Mistakes

Here are some common mistakes to avoid when simplifying radicals:

  • Not factoring the number under the radical: Failing to factor the number under the radical can make it difficult to simplify the expression.
  • Not applying the product property: Failing to apply the product property can make it difficult to separate the factors under the radical.
  • Not simplifying the square root of the number: Failing to simplify the square root of the number can make it difficult to simplify the expression.
  • Not applying the power property: Failing to apply the power property can make it difficult to simplify the square root of the variable.
  • Not combining the simplified factors: Failing to combine the simplified factors can make it difficult to simplify the expression.

Final Answer

The final answer is 5x6x\boxed{5 x^6 \sqrt{x}}.