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

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Understanding the Problem

When dealing with mathematical expressions involving square roots, it's essential to understand the properties of radicals and how they interact with each other. In this case, we're given the expression 1218\sqrt{12} \sqrt{18} and asked to simplify it. To do this, we need to apply the properties of radicals, specifically the property that states ab=ab\sqrt{a} \sqrt{b} = \sqrt{ab}.

Breaking Down the Expression

Let's start by breaking down the given expression into its individual components. We have 12\sqrt{12} and 18\sqrt{18}. To simplify these expressions, we need to find the prime factorization of the numbers inside the square roots.

Prime Factorization

The prime factorization of 12 is 22⋅32^2 \cdot 3, and the prime factorization of 18 is 2⋅322 \cdot 3^2. Now that we have the prime factorizations, we can rewrite the original expression as 22⋅32⋅32\sqrt{2^2 \cdot 3} \sqrt{2 \cdot 3^2}.

Applying the Property of Radicals

Now that we have the prime factorizations, we can apply the property of radicals that states ab=ab\sqrt{a} \sqrt{b} = \sqrt{ab}. In this case, we have 22⋅32⋅32=(22⋅3)(2⋅32)\sqrt{2^2 \cdot 3} \sqrt{2 \cdot 3^2} = \sqrt{(2^2 \cdot 3)(2 \cdot 3^2)}.

Simplifying the Expression

To simplify the expression, we need to multiply the numbers inside the square root. This gives us (22⋅3)(2⋅32)=22⋅3⋅2⋅32=23⋅33\sqrt{(2^2 \cdot 3)(2 \cdot 3^2)} = \sqrt{2^2 \cdot 3 \cdot 2 \cdot 3^2} = \sqrt{2^3 \cdot 3^3}.

Final Simplification

Now that we have the expression in the form 23⋅33\sqrt{2^3 \cdot 3^3}, we can simplify it further by taking the cube root of the numbers inside the square root. This gives us 23⋅33=2⋅32⋅3=66\sqrt{2^3 \cdot 3^3} = 2 \cdot 3 \sqrt{2 \cdot 3} = 6 \sqrt{6}.

Conclusion

In conclusion, the simplified form of the expression 1218\sqrt{12} \sqrt{18} is 666 \sqrt{6}. This is the correct answer among the options provided.

Comparison with Other Options

Let's compare our answer with the other options provided:

  • Option A: 1218=66\sqrt{12} \sqrt{18} = 6 \sqrt{6} (our answer)
  • Option B: 30=2⋅3⋅5=235\sqrt{30} = \sqrt{2 \cdot 3 \cdot 5} = \sqrt{2} \sqrt{3} \sqrt{5}
  • Option C: 56=52⋅3=5235 \sqrt{6} = 5 \sqrt{2 \cdot 3} = 5 \sqrt{2} \sqrt{3}
  • Option D: 65=656 \sqrt{5} = 6 \sqrt{5}
  • Option E: 66=62⋅3=6236 \sqrt{6} = 6 \sqrt{2 \cdot 3} = 6 \sqrt{2} \sqrt{3}

Final Answer

Based on our calculations, the correct answer is:

A. 1218\sqrt{12} \sqrt{18}

This is the simplified form of the given expression, and it matches one of the options provided.

Additional Tips and Tricks

When dealing with mathematical expressions involving square roots, it's essential to understand the properties of radicals and how they interact with each other. Here are some additional tips and tricks to help you simplify expressions involving square roots:

  • Use the property of radicals: The property of radicals states that ab=ab\sqrt{a} \sqrt{b} = \sqrt{ab}. This can help you simplify expressions involving multiple square roots.
  • Find the prime factorization: Finding the prime factorization of the numbers inside the square roots can help you simplify the expression.
  • Multiply the numbers inside the square root: When simplifying an expression involving multiple square roots, you can multiply the numbers inside the square root to simplify the expression.
  • Take the cube root: When simplifying an expression involving a square root, you can take the cube root of the numbers inside the square root to simplify the expression.

By following these tips and tricks, you can simplify expressions involving square roots and arrive at the correct answer.

Q: What is the property of radicals that states ab=ab\sqrt{a} \sqrt{b} = \sqrt{ab}?

A: The property of radicals that states ab=ab\sqrt{a} \sqrt{b} = \sqrt{ab} is a fundamental concept in mathematics that allows us to simplify expressions involving multiple square roots. This property states that the product of two square roots is equal to the square root of the product of the numbers inside the square roots.

Q: How do I find the prime factorization of a number?

A: To find the prime factorization of a number, you need to break down the number into its prime factors. For example, the prime factorization of 12 is 22⋅32^2 \cdot 3, and the prime factorization of 18 is 2⋅322 \cdot 3^2. You can find the prime factorization of a number by dividing it by prime numbers until you reach 1.

Q: How do I multiply the numbers inside the square root?

A: To multiply the numbers inside the square root, you simply multiply the numbers together. For example, if you have 22⋅32⋅32\sqrt{2^2 \cdot 3} \sqrt{2 \cdot 3^2}, you can multiply the numbers inside the square root to get (22⋅3)(2⋅32)\sqrt{(2^2 \cdot 3)(2 \cdot 3^2)}.

Q: What is the cube root of a number?

A: The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3, because 3⋅3⋅3=273 \cdot 3 \cdot 3 = 27.

Q: How do I take the cube root of a number inside a square root?

A: To take the cube root of a number inside a square root, you can simply take the cube root of the number and then multiply it by the square root of the remaining factors. For example, if you have 23⋅33\sqrt{2^3 \cdot 3^3}, you can take the cube root of the number to get 2⋅32⋅32 \cdot 3 \sqrt{2 \cdot 3}.

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

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

  • Not using the property of radicals to simplify expressions involving multiple square roots
  • Not finding the prime factorization of the numbers inside the square roots
  • Not multiplying the numbers inside the square root correctly
  • Not taking the cube root of the numbers inside the square root correctly

Q: How do I check my work when simplifying expressions involving square roots?

A: To check your work when simplifying expressions involving square roots, you can:

  • Use the property of radicals to simplify the expression
  • Find the prime factorization of the numbers inside the square roots
  • Multiply the numbers inside the square root correctly
  • Take the cube root of the numbers inside the square root correctly
  • Check that the final answer is in the simplest form possible

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

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

  • Calculating the area and perimeter of shapes with square roots
  • Calculating the volume of shapes with square roots
  • Calculating the distance and speed of objects with square roots
  • Calculating the area and volume of complex shapes with square roots

By following these tips and tricks, you can simplify expressions involving square roots and arrive at the correct answer.