1. Adding and Subtracting Radicals

Question

Definition: Radicals are expressions that include roots (like square roots). You can only add or subtract radicals that have the same index and radicand (the number inside the root).
Example:

Note: Non-like radicals cannot be combined, e.g.,

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2. Multiplying and Dividing Radicals
Multiplying Radicals: To multiply radicals, you multiply the coefficients and the radicands separately.
Example:

Dividing Radicals: To divide radicals, divide the coefficients and the radicands separately.
Example:

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3. Evaluating Roots
Evaluating Square Roots: The square root of a number is a value that, when multiplied by itself, gives that number.
Example:

Higher Roots:

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4. Products of Irrational Square Roots
When multiplying two irrational square roots, you may be able to simplify them.
Example:

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5. Identifying Roots
Identify roots by recognizing perfect squares or cubes:
• Perfect squares:
• Perfect cubes:
Example:
• Identify if 36 is a perfect square: Yes, since 6 times 6 = 36
• Identify if 50 is a perfect square: No.
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6. Estimating Values of Expressions with π (Pi)
Using Approximation: The value of π is approximately \( 3.14 \). Use this to estimate expressions involving π.
Example:
• Estimate the circumference of a circle with a radius of 5:
•
• Estimate the area of a circle with a radius of 3:
•
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please fill in all the qwestions that are not answered and keep the same layout

Answer 1

1. Adding and Subtracting Radicals
   Definition: Radicals are expressions that include roots (like square roots). You can only add or subtract radicals that have the same index and radicand (the number inside the root).
   Example:

• √2 + √2 = 2√2
• 3√5 - √5 = 2√5

Note: Non-like radicals cannot be combined, e.g.,

• √3 + √2 cannot be simplified further.

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2. Multiplying and Dividing Radicals
   Multiplying Radicals: To multiply radicals, you multiply the coefficients and the radicands separately.
   Example:

• √3 * √5 = √(3*5) = √15
• 2√2 * 3√2 = (23)(√2√2) = 6*2 = 12

Dividing Radicals: To divide radicals, divide the coefficients and the radicands separately.
Example:

• √8 / √2 = √(8/2) = √4 = 2
• 4√5 / 2√5 = (4/2)(√5/√5) = 2*1 = 2

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3. Evaluating Roots
   Evaluating Square Roots: The square root of a number is a value that, when multiplied by itself, gives that number.
   Example:

• √16 = 4
• √25 = 5

Higher Roots:

• The cube root of a number is a value that, when multiplied by itself three times, gives that number. For example,
  • ∛27 = 3 (since 3 * 3 * 3 = 27)
• The fourth root of a number is a value that, when multiplied by itself four times, gives that number. For example,
  • ∜16 = 2 (since 2 * 2 * 2 * 2 = 16)

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4. Products of Irrational Square Roots
   When multiplying two irrational square roots, you may be able to simplify them.
   Example:

• √2 * √8 = √(2*8) = √16 = 4
• √3 * √12 = √(3*12) = √36 = 6

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5. Identifying Roots
   Identify roots by recognizing perfect squares or cubes:
   • Perfect squares: Numbers like 1, 4, 9, 16, 25, etc., which are squares of integers (1², 2², 3², etc.).
   • Perfect cubes: Numbers like 1, 8, 27, 64, 125, etc., which are cubes of integers (1³, 2³, 3³, etc.).
   Example:
   • Identify if 36 is a perfect square: Yes, since 6 times 6 = 36
   • Identify if 50 is a perfect square: No.
   • Identify if 64 is a perfect cube: Yes, since 4 times 4 times 4 = 64
   • Identify if 30 is a perfect cube: No.

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6. Estimating Values of Expressions with π (Pi)
   Using Approximation: The value of π is approximately \( 3.14 \). Use this to estimate expressions involving π.
   Example:
   • Estimate the circumference of a circle with a radius of 5:

• Circumference = 2πr = 2 * 3.14 * 5 ≈ 31.4
  • Estimate the area of a circle with a radius of 3:
• Area = πr² = 3.14 * (3²) = 3.14 * 9 ≈ 28.26

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