What Is The Following Product?${ \sqrt{12} \cdot \sqrt{18} }$A. { \sqrt{30}$}$B. ${ 5 \sqrt{6}\$} C. ${ 6 \sqrt{5}\$} D. ${ 6 \sqrt{6}\$}

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Understanding the Problem

The given problem involves the multiplication of two square roots, 12\sqrt{12} and 18\sqrt{18}. To solve this, we need to apply the properties of square roots and simplify the expression.

Properties of Square Roots

Before we proceed, let's recall some important properties of square roots:

  • The square root of a number is a value that, when multiplied by itself, gives the original number.
  • The square root of a product is equal to the product of the square roots: aâ‹…b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}.
  • The square root of a quotient is equal to the quotient of the square roots: ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}.

Simplifying the Expression

Now, let's simplify the given expression using the properties of square roots:

12â‹…18=12â‹…18\sqrt{12} \cdot \sqrt{18} = \sqrt{12 \cdot 18}

Using the property of the product of square roots, we can rewrite the expression as:

12â‹…18=216\sqrt{12 \cdot 18} = \sqrt{216}

Simplifying the Square Root

To simplify the square root of 216, we need to find the largest perfect square that divides 216. In this case, 216 can be factored as:

216=36â‹…6216 = 36 \cdot 6

Since 36 is a perfect square, we can rewrite the expression as:

216=36â‹…6=36â‹…6=66\sqrt{216} = \sqrt{36 \cdot 6} = \sqrt{36} \cdot \sqrt{6} = 6\sqrt{6}

Conclusion

Therefore, the product of 12\sqrt{12} and 18\sqrt{18} is 666\sqrt{6}.

Answer

The correct answer is:

  • D. 666\sqrt{6}

Why is this the Correct Answer?

The correct answer is 666\sqrt{6} because we simplified the expression using the properties of square roots and factored the number 216 into its prime factors. This allowed us to rewrite the expression as the product of a perfect square and a square root, which we can simplify further.

What is the Importance of Simplifying Square Roots?

Simplifying square roots is an important skill in mathematics because it allows us to:

  • Reduce complex expressions: By simplifying square roots, we can reduce complex expressions into simpler ones, making it easier to work with them.
  • Identify perfect squares: Simplifying square roots helps us identify perfect squares, which is an important concept in mathematics.
  • Apply mathematical operations: Simplifying square roots allows us to apply mathematical operations, such as multiplication and division, more easily.

Real-World Applications of Simplifying Square Roots

Simplifying square roots has many real-world applications, including:

  • Engineering: Simplifying square roots is used in engineering to calculate distances, velocities, and accelerations.
  • Physics: Simplifying square roots is used in physics to calculate energies, forces, and momenta.
  • Computer Science: Simplifying square roots is used in computer science to optimize algorithms and data structures.

Conclusion

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

A: A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. A square, on the other hand, is the result of multiplying a number by itself. For example, the square of 4 is 16.

Q: How do I simplify a square root?

A: To simplify a square root, you need to find the largest perfect square that divides the number inside the square root. You can then rewrite the expression as the product of the perfect square and the square root of the remaining number.

Q: What is the property of the product of square roots?

A: The property of the product of square roots states that the square root of a product is equal to the product of the square roots. In other words, aâ‹…b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}.

Q: What is the property of the quotient of square roots?

A: The property of the quotient of square roots states that the square root of a quotient is equal to the quotient of the square roots. In other words, ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}.

Q: How do I simplify a square root of a fraction?

A: To simplify a square root of a fraction, you need to find the largest perfect square that divides the numerator and the denominator. You can then rewrite the expression as the product of the perfect square and the square root of the remaining fraction.

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

A: A rational number is a number that can be expressed as the ratio of two integers. For example, 3/4 is a rational number. An irrational number, on the other hand, is a number that cannot be expressed as the ratio of two integers. For example, the square root of 2 is an irrational number.

Q: How do I determine if a number is rational or irrational?

A: To determine if a number is rational or irrational, you need to check if it can be expressed as the ratio of two integers. If it can, then it is a rational number. If it cannot, then it is an irrational number.

Q: What are some real-world applications of square roots?

A: Square roots have many real-world applications, including:

  • Engineering: Square roots are used in engineering to calculate distances, velocities, and accelerations.
  • Physics: Square roots are used in physics to calculate energies, forces, and momenta.
  • Computer Science: Square roots are used in computer science to optimize algorithms and data structures.

Q: How do I use a calculator to simplify a square root?

A: To use a calculator to simplify a square root, you need to enter the expression into the calculator and press the square root button. The calculator will then display the simplified expression.

Q: What are some common mistakes to avoid when simplifying square roots?

A: Some common mistakes to avoid when simplifying square roots include:

  • Not finding the largest perfect square: Make sure to find the largest perfect square that divides the number inside the square root.
  • Not rewriting the expression correctly: Make sure to rewrite the expression as the product of the perfect square and the square root of the remaining number.
  • Not checking for rational or irrational numbers: Make sure to check if the number is rational or irrational before simplifying the square root.