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Simplify each square root expression. Describe the simplified form of the expression as rational or irrational. In your final answer, include all of your work.

1. square root of 121
2. square root of 48

3 answers

Let's simplify each square root expression step by step.

1. Square root of 121

To simplify \(\sqrt{121}\):

  1. Identify perfect squares: 121 is a perfect square because \(121 = 11 \times 11\).

  2. Take the square root: \[ \sqrt{121} = 11 \]

Thus, the simplified form of \(\sqrt{121}\) is 11, which is a rational number.

2. Square root of 48

To simplify \(\sqrt{48}\):

  1. Factor the number: First, we can factor 48 into its prime factors: \[ 48 = 16 \times 3 = 4 \times 4 \times 3 = 2^4 \times 3 \]

  2. Identify perfect squares: The largest perfect square that divides 48 is 16: \[ 48 = 16 \times 3 \]

  3. Take the square root: We can now simplify \(\sqrt{48}\): \[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \]

Thus, the simplified form of \(\sqrt{48}\) is \(4\sqrt{3}\). The number \(\sqrt{3}\) is an irrational number, so \(4\sqrt{3}\) is also irrational.

Summary of Results:

  1. \(\sqrt{121} = 11\) (rational)
  2. \(\sqrt{48} = 4\sqrt{3}\) (irrational)
can you do it without all the symbols and slashes

Of course! Let's simplify the square root expressions step by step without using symbols and slashes.

1. Square root of 121

  1. Recognize Perfect Square: The number 121 can be recognized as a perfect square because it is equal to 11 multiplied by 11.
  2. Calculate the Square Root: The square root of 121 is equal to 11.

The simplified form of the square root of 121 is 11, which is a rational number.

2. Square root of 48

  1. Factor the Number: To simplify the square root of 48, we can break it down into factors: 48 can be expressed as 16 multiplied by 3.
  2. Identify Perfect Squares: Here, 16 is a perfect square since it is equal to 4 multiplied by 4.
  3. Calculate the Square Root: The square root of 16 is 4, and so we can express the square root of 48 as the square root of 16 multiplied by the square root of 3. This gives us 4 multiplied by the square root of 3.

The simplified form of the square root of 48 is 4 times the square root of 3. Since the square root of 3 is not a whole number, this means the entire expression is irrational.

Summary of Results:

  1. The square root of 121 simplifies to 11 (a rational number).
  2. The square root of 48 simplifies to 4 times the square root of 3 (an irrational number).