Tier 3: Exponents & Roots

🧠 Key Laws of Exponents:

• \( a^m \cdot a^n = a^{m+n} \)
• \( (a^m)^n = a^{mn} \)
• \( (ab)^n = a^n \cdot b^n \)
• \( \frac{a^m}{a^n} = a^{m-n} \), where \( a \ne 0 \)

✏️ Example:

\( x^3 \cdot x^2 = x^5 \)

\( (2^3)^2 = 2^6 = 64 \)

✅ Try This:

1. \( x^4 \cdot x^3 \) → Answer: \( x^7 \)
2. \( (a^2)^3 \) → Answer: \( a^6 \)
3. \( (2x)^3 \) → Answer: \( 8x^3 \)

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🧠 Zero Exponent Rule

Any non-zero number raised to the power of zero equals 1:

\[ a^0 = 1 \quad \text{(for } a \ne 0\text{)} \]

✏️ Examples:

• \( 5^0 = 1 \)
• \( (-3)^0 = 1 \)
• \( (x^2)^0 = 1 \)

🧠 Negative Exponent Rule

A negative exponent means you take the reciprocal:

\[ a^{-n} = \frac{1}{a^n} \]

✏️ Examples:

• \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \)
• \( x^{-4} = \frac{1}{x^4} \)
• \( \left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \)

✅ Try These

1. \( 10^0 \) → Answer: 1
2. \( x^{-2} \) → Answer: \( \frac{1}{x^2} \)
3. \( \left(\frac{5}{2}\right)^{-1} \) → Answer: \( \frac{2}{5} \)

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🧠 Square and Cube Roots

The square root of a number \( x \) is a number \( y \) such that \( y^2 = x \).

The cube root of a number \( x \) is a number \( y \) such that \( y^3 = x \).

✏️ Examples:

• \( \sqrt{16} = 4 \) because \( 4^2 = 16 \)
• \( \sqrt{100} = 10 \)
• \( \sqrt[3]{27} = 3 \) because \( 3^3 = 27 \)
• \( \sqrt[3]{-8} = -2 \) because \( (-2)^3 = -8 \)

✅ Try These

1. \( \sqrt{81} \) → Answer: 9
2. \( \sqrt[3]{64} \) → Answer: 4
3. \( \sqrt[3]{-27} \) → Answer: -3

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🧠 What Are Rational Exponents?

Rational exponents are another way to write roots using powers. The denominator tells the root, and the numerator tells the power:

\[ a^{1/n} = \sqrt[n]{a}, \quad a^{m/n} = \sqrt[n]{a^m} \]

✏️ Examples:

• \( 8^{1/3} = \sqrt[3]{8} = 2 \)
• \( 16^{1/2} = \sqrt{16} = 4 \)
• \( 27^{2/3} = \sqrt[3]{27^2} = \sqrt[3]{729} = 9 \)
• \( 81^{3/4} = \sqrt[4]{81^3} = \sqrt[4]{531441} = 27 \)

✅ Try These

1. \( 64^{1/2} \) → Answer: 8
2. \( 125^{2/3} \) → Answer: 25
3. \( 100^{3/2} \) → Answer: 1000

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