Square Root: A number which when multiplied by itself gives the original number

Symbol: \(\sqrt{~}\) (radical sign)

Formula: If \(\sqrt{n} = a\), then \(a \times a = n\) or \(a^2 = n\)

Perfect Square: A number whose square root is a whole number

Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...

Step 1: Find two perfect squares between which the number lies

Step 2: Identify the square roots of these perfect squares

Step 3: The square root of the number lies between these two values

Step 4: Estimate closer value based on proximity

Estimate \(\sqrt{50}\):

Step 1: \(49 < 50 < 64\) (perfect squares)

Step 2: \(\sqrt{49} = 7\) and \(\sqrt{64} = 8\)

Step 3: Therefore, \(7 < \sqrt{50} < 8\)

Answer: \(\sqrt{50} \approx 7.1\) (closer to 7)

\(\sqrt{20}\): Between \(\sqrt{16} = 4\) and \(\sqrt{25} = 5\), so \(\sqrt{20} \approx 4.5\)

\(\sqrt{75}\): Between \(\sqrt{64} = 8\) and \(\sqrt{81} = 9\), so \(\sqrt{75} \approx 8.7\)

Every positive number has TWO square roots: one positive and one negative

Formula: If \(x^2 = n\), then \(x = \pm\sqrt{n}\)

\(\sqrt{25} = 5\) (principal/positive square root only)

\(-\sqrt{25} = -5\) (negative square root)

\(\pm\sqrt{25} = \pm 5\) (both positive and negative)

Both \(5^2 = 25\) and \((-5)^2 = 25\)

Square roots of 36: \(\pm\sqrt{36} = \pm 6\) (both 6 and -6)

Square roots of 64: \(\pm\sqrt{64} = \pm 8\) (both 8 and -8)

Square roots of 100: \(\pm\sqrt{100} = \pm 10\) (both 10 and -10)

✓ The square root of a positive number exists

✗ The square root of a negative number is NOT a real number

Example: \(\sqrt{-16}\) is not a real number

Step 1: Estimate the positive square root first (as in Section 2)

Step 2: The negative square root is the opposite of the positive

If \(\sqrt{n} \approx a\), then \(-\sqrt{n} \approx -a\)

Estimate \(\pm\sqrt{30}\):
Since \(25 < 30 < 36\), we have \(5 < \sqrt{30} < 6\)
Estimate: \(\sqrt{30} \approx 5.5\) and \(-\sqrt{30} \approx -5.5\)
So \(\pm\sqrt{30} \approx \pm 5.5\)

Estimate \(\pm\sqrt{85}\):
Since \(81 < 85 < 100\), we have \(9 < \sqrt{85} < 10\)
Estimate: \(\pm\sqrt{85} \approx \pm 9.2\)

Squaring and taking square root are INVERSE operations

They "undo" each other

Formula 1: \((\sqrt{n})^2 = n\)

Example: \((\sqrt{16})^2 = 4^2 = 16\)

Formula 2: \(\sqrt{n^2} = |n|\) (absolute value)

Example: \(\sqrt{5^2} = \sqrt{25} = 5\)

Example: \(\sqrt{(-5)^2} = \sqrt{25} = 5\) (not -5!)

Product Property: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)

Example: \(\sqrt{36} = \sqrt{4 \times 9} = \sqrt{4} \times \sqrt{9} = 2 \times 3 = 6\)

Quotient Property: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)

Example: \(\sqrt{\frac{64}{16}} = \frac{\sqrt{64}}{\sqrt{16}} = \frac{8}{4} = 2\)

Equation Type: \(x^2 = n\)

Solution: \(x = \pm\sqrt{n}\)

Remember to include BOTH positive and negative solutions!

Step 1: Isolate \(x^2\) on one side of the equation

Step 2: Take square root of both sides

Step 3: Write both positive and negative solutions

Step 4: Simplify if possible

Example 1: \(x^2 = 49\)
Take square root: \(x = \pm\sqrt{49}\)
Solution: \(x = 7\) or \(x = -7\)

Example 2: \(x^2 - 25 = 0\)
Isolate: \(x^2 = 25\)
Take square root: \(x = \pm\sqrt{25}\)
Solution: \(x = 5\) or \(x = -5\)

Example 3: \(3x^2 = 75\)
Divide by 3: \(x^2 = 25\)
Take square root: \(x = \pm 5\)

Cube Root: A number which when multiplied by itself THREE times gives the original number

Symbol: \(\sqrt[3]{~}\) or \(\sqrt[3]{~}\)

Formula: If \(\sqrt[3]{n} = a\), then \(a \times a \times a = n\) or \(a^3 = n\)

Perfect Cube: A number whose cube root is a whole number

Examples: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000...

Cube roots of negative numbers ARE real numbers!

Unlike square roots, cube roots can be negative

Every number has only ONE real cube root (not two like square roots)

Rule: The cube root has the SAME SIGN as the original number

✓ Cube root of positive → positive

✓ Cube root of negative → negative

Positive Cubes:

\(\sqrt[3]{64} = 4\) because \(4^3 = 64\)

\(\sqrt[3]{125} = 5\) because \(5^3 = 125\)

Negative Cubes:

\(\sqrt[3]{-8} = -2\) because \((-2)^3 = -8\)

\(\sqrt[3]{-27} = -3\) because \((-3)^3 = -27\)

\(\sqrt[3]{-125} = -5\) because \((-5)^3 = -125\)

Equation Type: \(x^3 = n\)

Solution: \(x = \sqrt[3]{n}\)

Only ONE solution (not ± like square roots!)

Step 1: Isolate \(x^3\) on one side of the equation

Step 2: Take cube root of both sides

Step 3: Simplify the answer

Example 1: \(x^3 = 64\)
Take cube root: \(x = \sqrt[3]{64}\)
Solution: \(x = 4\)

Example 2: \(x^3 = -27\)
Take cube root: \(x = \sqrt[3]{-27}\)
Solution: \(x = -3\)

Example 3: \(2x^3 = 250\)
Divide by 2: \(x^3 = 125\)
Take cube root: \(x = \sqrt[3]{125}\)
Solution: \(x = 5\)

Step 1: Find two perfect cubes between which the number lies

Step 2: Identify the cube roots of these perfect cubes

Step 3: The cube root of the number lies between these two values

Step 4: Estimate closer value based on proximity

Estimate \(\sqrt[3]{100}\):

Step 1: \(64 < 100 < 125\) (perfect cubes)

Step 2: \(\sqrt[3]{64} = 4\) and \(\sqrt[3]{125} = 5\)

Step 3: Therefore, \(4 < \sqrt[3]{100} < 5\)

Answer: \(\sqrt[3]{100} \approx 4.6\) (closer to 5)

\(\sqrt[3]{50}\): Between \(\sqrt[3]{27} = 3\) and \(\sqrt[3]{64} = 4\), so \(\sqrt[3]{50} \approx 3.7\)

\(\sqrt[3]{-50}\): Between \(\sqrt[3]{-27} = -3\) and \(\sqrt[3]{-64} = -4\), so \(\sqrt[3]{-50} \approx -3.7\)

\(\sqrt[3]{300}\): Between \(\sqrt[3]{216} = 6\) and \(\sqrt[3]{343} = 7\), so \(\sqrt[3]{300} \approx 6.7\)

⚡ Remember: Square roots have ± | Cube roots can be negative | Estimation uses perfect squares/cubes! ⚡

1. Product Property: \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\) and \(\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}\)

2. Quotient Property: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\) and \(\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}\)

3. Power Property: \((\sqrt{a})^2 = a\) and \((\sqrt[3]{a})^3 = a\)

4. Inverse Property: \(\sqrt{a^2} = |a|\) and \(\sqrt[3]{a^3} = a\)

✓ Memorize perfect squares and cubes: At least 1-12 for squares and 1-10 for cubes

✓ Practice estimation: Identify which two perfect squares/cubes a number falls between

✓ Remember the difference: Square roots have ±, cube roots don't

✓ Negative numbers: Square roots don't exist (real), cube roots do exist

✓ Check your work: Square or cube your answer to verify

✓ Use properties: Break down large numbers using product/quotient properties