{"id":9900,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9900"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"laws-of-exponent","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/laws-of-exponent\/","title":{"rendered":"Laws Of Exponent"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Exponents & Powers<\/strong><\/h2>\n

Chapter: <\/strong>Laws Of Exponents<\/strong><\/h3>\n

Reference: – Review of Exponents, Law 1: Product Rule (a^m \u00d7 a^n = a^(m+n)), Law 2: Quotient Rule (a^m \u00f7 a^n = a^(m-n)), Law 3: Power of a Power ((a^m)^n = a^(m\u00d7n)), Law 4: Power of a Product ((ab)^m = a^m b^m), Law 5: Power of a Quotient ((a\/b)^m = a^m\/b^m), Zero Exponent Rule (a\u2070 = 1), Negative Exponent Rule (a^(-n) = 1\/a^n), Combining Multiple Laws, Solved Examples, Odd-One-Out Problems, Common Mistakes<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • All Seven Laws of Exponents<\/em><\/li>\n
  • How to Apply Multiple Laws Together<\/em><\/li>\n
  • When to Use Each Law<\/em><\/li>\n
  • Simplify Expressions Using Laws of Exponents<\/em><\/li>\n<\/ul>\n

    Introduction to Laws of Exponents<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    The laws of exponents are rules that tell us how to simplify expressions involving powers (exponents). These rules apply when the bases are the same (for multiplication and division) or when we raise a power to another power, a product to a power, or a quotient to a power.<\/p>\n

    When we use the laws of exponents, we essentially ask:<\/p>\n

    “Which rule applies here? How can I simplify this expression?”<\/p>\n

    Once we master these laws, we can simplify complex exponential expressions quickly and correctly.<\/p>\n

    Importance of Laws of Exponents<\/u><\/strong><\/p>\n

      \n
    • Essential for algebra, calculus, and higher mathematics<\/li>\n
    • Used in scientific notation (working with very large\/small numbers)<\/li>\n
    • Helps simplify expressions in physics, chemistry, and engineering<\/li>\n
    • Foundation for exponential growth and decay problems<\/li>\n<\/ul>\n

      Example<\/strong><\/p>\n

      Using laws of exponents: 2\u00b3 \u00d7 2\u2074 = 2\u2077, 2\u2077 \u00f7 2\u00b3 = 2\u2074, (2\u00b3)\u2074 = 2\u00b9\u00b2, (2x)\u00b3 = 8x\u00b3<\/p>\n

      Subtopics<\/u><\/strong><\/p>\n

      1. Law 1 \u2013 Product Rule (Same Base)<\/strong><\/p>\n

      When multiplying two powers with the same base, add the exponents.<\/p>\n

      Formula:\u00a0a^m \u00d7 a^n = a^(m + n)<\/p>\n

      Example 1:\u00a03\u2075 \u00d7 3\u00b2 = 3^(5+2) = 3\u2077<\/p>\n

      Example 2:\u00a0x\u2074 \u00d7 x\u2076 = x^(4+6) = x\u00b9\u2070<\/p>\n

      Example 3:\u00a02 \u00d7 2\u00b3 = 2\u00b9 \u00d7 2\u00b3 = 2^(1+3) = 2\u2074 = 16<\/p>\n

      Example 4:\u00a05\u00b2 \u00d7 5\u00b3 \u00d7 5\u2074 = 5^(2+3+4) = 5\u2079<\/p>\n

      2. Law 2 \u2013 Quotient Rule (Same Base)<\/strong><\/p>\n

      When dividing two powers with the same base, subtract the exponents (numerator exponent minus denominator exponent).<\/p>\n

      Formula:\u00a0a^m \u00f7 a^n = a^(m – n) (a \u2260 0)<\/p>\n

      Example 1:\u00a07\u2076 \u00f7 7\u00b2 = 7^(6-2) = 7\u2074<\/p>\n

      Example 2:\u00a0x\u2079 \u00f7 x\u2074 = x^(9-4) = x\u2075<\/p>\n

      Example 3:\u00a010\u2075 \u00f7 10\u00b3 = 10^(5-3) = 10\u00b2 = 100<\/p>\n

      Example 4:\u00a02\u2074 \u00f7 2\u2074 = 2^(4-4) = 2\u2070 = 1<\/p>\n

      3. Law 3 \u2013 Power of a Power<\/strong><\/p>\n

      When raising a power to another power, multiply the exponents.<\/p>\n

      Formula:\u00a0(a^m)^n = a^(m \u00d7 n)<\/p>\n

      Example 1:\u00a0(2\u00b3)\u2074 = 2^(3\u00d74) = 2\u00b9\u00b2<\/p>\n

      Example 2:\u00a0(x\u00b2)\u2075 = x^(2\u00d75) = x\u00b9\u2070<\/p>\n

      Example 3:\u00a0(5\u00b2)\u00b3 = 5^(2\u00d73) = 5\u2076 = 15625<\/p>\n

      Example 4:\u00a0[(3\u00b2)\u00b3]\u2074 = 3^(2\u00d73\u00d74) = 3\u00b2\u2074<\/p>\n

      4. Law 4 \u2013 Power of a Product<\/strong><\/p>\n

      When raising a product to a power, raise each factor to that power.<\/p>\n

      Formula:\u00a0(ab)^m = a^m \u00d7 b^m<\/p>\n

      Example 1:\u00a0(2 \u00d7 5)\u00b3 = 2\u00b3 \u00d7 5\u00b3 = 8 \u00d7 125 = 1000 (Check: 10\u00b3 = 1000)<\/p>\n

      Example 2:\u00a0(3x)\u2074 = 3\u2074 \u00d7 x\u2074 = 81x\u2074<\/p>\n

      Example 3:\u00a0(2y)\u2075 = 2\u2075 \u00d7 y\u2075 = 32y\u2075<\/p>\n

      Example 4:\u00a0(4ab)\u00b3 = 4\u00b3 \u00d7 a\u00b3 \u00d7 b\u00b3 = 64a\u00b3b\u00b3<\/p>\n

      5. Law 5 \u2013 Power of a Quotient<\/strong><\/p>\n

      When raising a quotient to a power, raise both numerator and denominator to that power.<\/p>\n

      Formula:\u00a0(a\/b)^m = a^m \/ b^m (b \u2260 0)<\/p>\n

      Example 1:\u00a0(3\/4)\u00b2 = 3\u00b2 \/ 4\u00b2 = 9\/16<\/p>\n

      Example 2:\u00a0(x\/y)\u00b3 = x\u00b3 \/ y\u00b3 (y \u2260 0)<\/p>\n

      Example 3:\u00a0(2x\/5)\u2074 = (2x)\u2074 \/ 5\u2074 = 16x\u2074 \/ 625<\/p>\n

      Example 4:\u00a0(5\/2)\u00b3 = 5\u00b3 \/ 2\u00b3 = 125\/8 = 15.625<\/p>\n

      6. Law 6 \u2013 Zero Exponent Rule<\/strong><\/p>\n

      Any non-zero number raised to the power zero equals 1.<\/p>\n

      Formula:\u00a0a\u2070 = 1 (a \u2260 0)<\/p>\n

      Example 1:\u00a07\u2070 = 1<\/p>\n

      Example 2:\u00a0100\u2070 = 1<\/p>\n

      Example 3:\u00a0x\u2070 = 1 (x \u2260 0)<\/p>\n

      Example 4:\u00a0(3x)\u2070 = 1 (3x \u2260 0)<\/p>\n

      Example 5:\u00a00\u2070 is undefined (not covered in Grade 8)<\/p>\n

      7. Law 7 \u2013 Negative Exponent Rule<\/strong><\/p>\n

      A negative exponent means take the reciprocal of the base raised to the positive exponent.<\/p>\n

      Formula:\u00a0a^(-n) = 1\/a^n (a \u2260 0)<\/p>\n

      Also, 1\/a^(-n) = a^n<\/p>\n

      Example 1:\u00a02\u207b\u00b3 = 1\/2\u00b3 = 1\/8<\/p>\n

      Example 2:\u00a05\u207b\u00b2 = 1\/5\u00b2 = 1\/25<\/p>\n

      Example 3:\u00a0x\u207b\u2074 = 1\/x\u2074<\/p>\n

      Example 4:\u00a0(3\/4)\u207b\u00b2 = (4\/3)\u00b2 = 16\/9<\/p>\n

      Example 5:\u00a01\/2\u207b\u00b3 = 2\u00b3 = 8<\/p>\n

      8. Combining Multiple Laws<\/strong><\/p>\n

      Often, we need to use more than one law to simplify an expression.<\/p>\n

      Example 1:<\/strong>\u00a0Simplify (2\u00b2 \u00d7 2\u00b3)\u2074<\/p>\n

      First, inside parentheses: 2\u00b2 \u00d7 2\u00b3 = 2^(2+3) = 2\u2075
      \nThen, (2\u2075)\u2074 = 2^(5\u00d74) = 2\u00b2\u2070<\/p>\n

      Example 2:<\/strong>\u00a0Simplify (3\u00b2)\u00b3 \u00d7 3\u2074<\/p>\n

      First, (3\u00b2)\u00b3 = 3^(2\u00d73) = 3\u2076
      \nThen, 3\u2076 \u00d7 3\u2074 = 3^(6+4) = 3\u00b9\u2070<\/p>\n

      Example 3:<\/strong>\u00a0Simplify (x\u00b3 \u00d7 x\u00b2)\u2074 \u00f7 x\u2076<\/p>\n

      Inside: x\u00b3 \u00d7 x\u00b2 = x\u2075
      \n(x\u2075)\u2074 = x\u00b2\u2070
      \nx\u00b2\u2070 \u00f7 x\u2076 = x^(20-6) = x\u00b9\u2074<\/p>\n

      Example 4:<\/strong>\u00a0Simplify (2\u207b\u00b2)\u00b3 \u00d7 2\u2075<\/p>\n

      (2\u207b\u00b2)\u00b3 = 2^(-2\u00d73) = 2\u207b\u2076
      \n2\u207b\u2076 \u00d7 2\u2075 = 2^(-6+5) = 2\u207b\u00b9 = 1\/2<\/p>\n

      Solved Examples<\/strong><\/p>\n

      Example 1 \u2013 Product Rule:<\/strong>\u00a0Simplify 4\u2077 \u00d7 4\u00b3<\/p>\n

      Solution:<\/strong>\u00a04^(7+3) = 4\u00b9\u2070<\/p>\n

      Answer:<\/strong>\u00a04\u00b9\u2070<\/p>\n

      \u00a0<\/p>\n

      Example 2 \u2013 Quotient Rule:<\/strong>\u00a0Simplify 9\u2078 \u00f7 9\u2075<\/p>\n

      Solution:<\/strong>\u00a09^(8-5) = 9\u00b3<\/p>\n

      Answer:<\/strong>\u00a09\u00b3<\/p>\n

      \u00a0<\/p>\n

      Example 3 \u2013 Power of a Power:<\/strong>\u00a0Simplify (5\u00b3)\u2074<\/p>\n

      Solution:<\/strong>\u00a05^(3\u00d74) = 5\u00b9\u00b2<\/p>\n

      Answer:<\/strong>\u00a05\u00b9\u00b2<\/p>\n

      \u00a0<\/p>\n

      Example 4 \u2013 Power of a Product:<\/strong>\u00a0Simplify (2x)\u2075<\/p>\n

      Solution:<\/strong>\u00a02\u2075 \u00d7 x\u2075 = 32x\u2075<\/p>\n

      Answer:<\/strong>\u00a032x\u2075<\/p>\n

      \u00a0<\/p>\n

      Example 5 \u2013 Power of a Quotient:<\/strong>\u00a0Simplify (3\/5)\u00b3<\/p>\n

      Solution:<\/strong>\u00a03\u00b3 \/ 5\u00b3 = 27\/125<\/p>\n

      Answer:<\/strong>\u00a027\/125<\/p>\n

      \u00a0<\/p>\n

      Example 6 \u2013 Combining Laws:<\/strong>\u00a0Simplify (2\u00b3 \u00d7 2\u00b2)\u00b3<\/p>\n

      Solution:<\/strong>\u00a0Inside: 2\u00b3 \u00d7 2\u00b2 = 2\u2075
      \n(2\u2075)\u00b3 = 2\u00b9\u2075<\/p>\n

      Answer:<\/strong>\u00a02\u00b9\u2075<\/p>\n

      \u00a0<\/p>\n

      Example 7 \u2013 Negative Exponent:<\/strong>\u00a0Simplify 3\u207b\u2074 \u00d7 3\u2076<\/p>\n

      Solution:<\/strong>\u00a03^(-4+6) = 3\u00b2 = 9<\/p>\n

      Answer:<\/strong>\u00a09<\/p>\n

      Common Mistakes to Avoid<\/u><\/strong><\/p>\n

      Mistake 1 \u2013 Adding exponents when bases are different<\/strong>
      \n2\u00b3 \u00d7 3\u00b2 cannot be simplified to 6\u2075 (wrong).
      \nCorrect understanding: Laws of exponents apply only to the same base.<\/p>\n

      Mistake 2 \u2013 Confusing product rule with power of a power<\/strong>
      \n(2\u00b3)\u2074 = 2\u00b9\u00b2, not 2\u2077.
      \nCorrect understanding: Power of a power multiplies exponents; product rule adds exponents.<\/p>\n

      Mistake 3 \u2013 Forgetting to raise the coefficient<\/strong>
      \n(3x)\u00b2 = 9x\u00b2, not 3x\u00b2.
      \nCorrect understanding: Raise EVERY factor inside parentheses to the power.<\/p>\n

      Mistake 4 \u2013 Misapplying negative exponent rule<\/strong>
      \n2\u207b\u00b3 = 1\/8, not -8. The negative exponent means reciprocal, not negative number.
      \nCorrect understanding: a^(-n) = 1\/a^n (positive denominator).<\/p>\n

      Mistake 5 \u2013 Thinking a\u2070 = 0<\/strong>
      \n5\u2070 = 1, not 0.
      \nCorrect understanding: Any non-zero number to the power zero equals 1.<\/p>\n

      Mistake 6 \u2013 Subtracting exponents in wrong order<\/strong>
      \na^m \u00f7 a^n = a^(m-n). Make sure denominator exponent is subtracted from numerator exponent.
      \nCorrect understanding: m – n, not n – m.<\/p>\n

      \u00a0<\/p>\n

      Quick Reference Summary \u2013 The Seven Laws<\/strong><\/p>\n

      \n\n\n\n\n\n\n\n\n\n\n\n
      \n

      Law Name<\/p>\n<\/td>\n

      		\n

      Formula<\/p>\n<\/td>\n

      		\n

      Example<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Product Rule<\/p>\n<\/td>\n

      		\n

      a^m \u00d7 a^n = a^(m+n)<\/p>\n<\/td>\n

      		\n

      3\u00b2 \u00d7 3\u2074 = 3\u2076<\/p>\n<\/td>\n<\/tr>\n

      \n

      Quotient Rule<\/p>\n<\/td>\n

      		\n

      a^m \u00f7 a^n = a^(m-n)<\/p>\n<\/td>\n

      		\n

      5\u2077 \u00f7 5\u00b3 = 5\u2074<\/p>\n<\/td>\n<\/tr>\n

      \n

      Power of a Power<\/p>\n<\/td>\n

      		\n

      (a^m)^n = a^(m\u00d7n)<\/p>\n<\/td>\n

      		\n

      (2\u00b3)\u2074 = 2\u00b9\u00b2<\/p>\n<\/td>\n<\/tr>\n

      \n

      Power of a Product<\/p>\n<\/td>\n

      		\n

      (ab)^m = a^m b^m<\/p>\n<\/td>\n

      		\n

      (3x)\u00b2 = 9x\u00b2<\/p>\n<\/td>\n<\/tr>\n

      \n

      Power of a Quotient<\/p>\n<\/td>\n

      		\n

      (a\/b)^m = a^m\/b^m<\/p>\n<\/td>\n

      		\n

      (2\/3)\u00b3 = 8\/27<\/p>\n<\/td>\n<\/tr>\n

      \n

      Zero Exponent<\/p>\n<\/td>\n

      		\n

      a\u2070 = 1 (a \u2260 0)<\/p>\n<\/td>\n

      		\n

      7\u2070 = 1<\/p>\n<\/td>\n<\/tr>\n

      \n

      Negative Exponent<\/p>\n<\/td>\n

      		\n

      a^(-n) = 1\/a^n<\/p>\n<\/td>\n

      		\n

      2\u207b\u00b3 = 1\/8<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Remember:<\/strong><\/p>\n

        \n
      • Same base for product\/quotient rules<\/li>\n
      • Multiply exponents for power of a power<\/li>\n
      • Distribute exponent to all factors inside parentheses<\/li>\n
      • 0\u2070 is undefined<\/li>\n<\/ul>\n

        \u00a0<\/p>\n

        <\/p>\n