Menyederhanakan Operasi Bilangan Berpangkat Yang Bilangan Pokoknya Tidak Sama

Main Ideas / Lesson

The video teaches how to simplify products, quotients, and expressions involving exponents when the bases are different by:

1. Rewriting the terms so they share the same base (using prime-factor/power forms of numbers).
2. Then applying exponent rules:
   • Multiplication (same base): (a^m \cdot a^n = a^{m+n})
   • Division (same base): (a^m / a^n = a^{m-n})
3. Using a consistent approach across all examples: Choose the larger composite base (non-prime) and rewrite it into a power of a smaller base, then proceed with the exponent rules.

Note: The subtitles sometimes say “non-prime,” “prime,” and “larger base” unclearly. The intended method is to rewrite composite bases as powers, for example: (8 = 2^3,\; 9 = 3^2,\; 6 = 2\cdot 3,\; 125 = 5^3.)

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Method (Step-by-Step)

For expressions like ( \text{(power)} \times \text{(power)} ) or ( \text{(power)} / \text{(power)} ) where the bases differ:

• Step 1: Identify the bases

  • Example forms: (a^m \cdot b^n), (a^m / b^n), etc.

• Step 2: Rewrite the larger/composite base into prime-factor/power form

  • Express it as:
    • a single power (e.g., (8 = 2^3), (9 = 3^2), (125 = 5^3)), or
    • a product of primes that can be converted to exponent form (e.g., (6 = 2\cdot 3)).

• Step 3: Apply the power-of-a-power rule

  • If (b = c^k), then ((c^k)^n = c^{k\cdot n}).

• Step 4: Convert the expression so bases match

  • After rewriting, group terms with the same base.

• Step 5: Use multiplication/division exponent rules

  • For the same base:
    • Multiply: add exponents
    • Divide: subtract exponents

• Step 6: Simplify to final exponent form

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Worked Examples (Main Outcomes)

Example 1

Expression: (2^4 \cdot 8^5)

• Rewrite: (8 = 2^3)
• Power-of-a-power: ((2^3)^5 = 2^{3\cdot 5} = 2^{15})
• Same base multiplication: (2^4 \cdot 2^{15} = 2^{4+15} = 2^{19})

Result: (\boxed{2^{19}})

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Example 2

Expression: (9^6 / 3^4)

• Rewrite: (9 = 3^2)
• Power-of-a-power: ((3^2)^6 = 3^{2\cdot 6} = 3^{12})
• Same base division: (3^{12} / 3^4 = 3^{12-4} = 3^8)

Result: (\boxed{3^{8}})

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Example 3

Expression: (6^7 \cdot 2^4 / 3^3)

• Rewrite: (6 = 2\cdot 3), so (6^7 = (2\cdot 3)^7 = 2^7\cdot 3^7)
• Substitute and rearrange:
  • (2^7 \cdot 2^4 = 2^{7+4} = 2^{11})
  • (3^7 / 3^3 = 3^{7-3} = 3^4)

Result: (\boxed{2^{11}\cdot 3^{4}})

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Example 4

Expression: (2^6 \cdot 125^3 / 5)

• Rewrite: (125 = 5^3)
• Power-of-a-power: ((5^3)^3 = 5^{9})
• Expression becomes: (2^6 \cdot 5^9 / 5^1)
• Same base division: (5^9 / 5^1 = 5^{9-1} = 5^8)

Result: (\boxed{2^{6}\cdot 5^{8}})

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Speakers / Sources

• Unidentified speaker / instructor (only person talking; no name provided in the subtitles).