Simplify $\sqrt{96}$.A. $24 \sqrt{4}$ B. $6 \sqrt{4}$ C. $4 \sqrt{6}$ D. $16 \sqrt{6}$

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Introduction

Simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. It involves expressing a square root in its simplest form, which can be achieved by factoring the number inside the square root sign. In this article, we will explore the process of simplifying square roots, using the example of 96\sqrt{96}.

Understanding Square Roots

A square root of a number 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. The square root of a number can be represented by the symbol \sqrt{}. When simplifying square roots, we aim to express the number inside the square root sign in its simplest form.

Simplifying 96\sqrt{96}

To simplify 96\sqrt{96}, we need to factor the number 96. We can start by breaking down 96 into its prime factors:

96 = 2 × 2 × 2 × 2 × 3

We can rewrite 96 as:

96 = 2^4 × 3

Now, we can take the square root of each factor:

96\sqrt{96} = 24×3\sqrt{2^4 × 3}

Using the property of square roots that a2=a\sqrt{a^2} = a, we can simplify the expression:

24×3\sqrt{2^4 × 3} = 2^2 × 3\sqrt{3}

96\sqrt{96} = 4 × 3\sqrt{3}

However, we can simplify further by recognizing that 4 is a perfect square:

96\sqrt{96} = 4 × 3\sqrt{3} = 43\sqrt{3}

But we can simplify it even more by recognizing that 4 is a perfect square of 2, and 3 is not a perfect square of 2.

Q&A: Simplifying Square Roots

Q: What is a square root?

A: A square root of a number 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.

Q: How do I simplify a square root?

A: To simplify a square root, you need to factor the number inside the square root sign. You can start by breaking down the number into its prime factors. Then, you can take the square root of each factor and simplify the expression.

Q: What is the difference between a perfect square and a non-perfect square?

A: A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be expressed as 4^2. A non-perfect square is a number that cannot be expressed as the square of an integer.

Q: How do I know if a number is a perfect square?

A: To determine if a number is a perfect square, you can try to find the square root of the number. If the square root is an integer, then the number is a perfect square.

Q: Can I simplify a square root with a variable?

A: Yes, you can simplify a square root with a variable. To do this, you need to factor the variable and then take the square root of each factor.

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 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: Can I simplify a square root with a rational number?

A: Yes, you can simplify a square root with a rational number. To do this, you need to factor the rational number and then take the square root of each factor.

Q: What is the difference between a real and an imaginary number?

A: A real number is a number that can be expressed as a rational or irrational number. For example, 3 and the square root of 2 are real numbers. An imaginary number is a number that cannot be expressed as a real number. For example, the square root of -1 is an imaginary number.

Q: Can I simplify a square root with an imaginary number?

A: Yes, you can simplify a square root with an imaginary number. To do this, you need to factor the imaginary number and then take the square root of each factor.

Conclusion

Simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. By understanding the properties of square roots and how to simplify them, you can solve a wide range of mathematical problems. Remember to factor the number inside the square root sign, take the square root of each factor, and simplify the expression.

Common Mistakes to Avoid

  • Not factoring the number inside the square root sign
  • Not taking the square root of each factor
  • Not simplifying the expression
  • Not checking if the number is a perfect square

Tips and Tricks

  • Use the property of square roots that a2=a\sqrt{a^2} = a to simplify expressions
  • Use the property of square roots that ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b} to simplify expressions
  • Use the property of square roots that ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} to simplify expressions
  • Check if the number is a perfect square before simplifying the expression

Practice Problems

  • Simplify 16\sqrt{16}
  • Simplify 25\sqrt{25}
  • Simplify 36\sqrt{36}
  • Simplify 49\sqrt{49}
  • Simplify 64\sqrt{64}

Answer Key

  • 16=4\sqrt{16} = 4
  • 25=5\sqrt{25} = 5
  • 36=6\sqrt{36} = 6
  • 49=7\sqrt{49} = 7
  • 64=8\sqrt{64} = 8