Math Laws of indices

• What is the first rule when multiplying powers with the same base?

  Add the powers together
• How do you express $3^2 \times 3^5$ as a power of 3?


  $3^7$
• What does $3^2$ equal in terms of multiplication?


  $3 \times 3$
• How many threes are multiplied together in $3^5$?


  Five threes
• How do you derive $3^7$ from $3^2 \times 3^5$?


  By adding the exponents: $2 +$$5 =$$7$
• What happens when the bases are different, such as $3^2 \times 2^5$?


  You cannot combine the powers
• What is the second rule when dividing powers with the same base?

  Subtract the powers
• How do you express $5^7 \div 5^3$ as a power of 5?


  $5^4$
• How do you arrive at $5^4$ from $5^7 \div 5^3$?


  By subtracting the exponents: $7 - 3 =$$4$
• What does $2^4\text{ }^3$ mean?


  It means $2^4$ multiplied by itself three times
• How do you express $2^4\text{ }^3$ as a power of 2?


  $2^{12}$
• How do you derive $2^{12}$ from $2^4\text{ }^3$?


  By multiplying the exponents: $4 \times 3 =$$12$
• What are the three rules for working with powers of the same base?

  • When multiplying, add the powers.
  • When dividing, subtract the powers.
  • When raising a power to another power, multiply the powers.
• What is the result of $3^4 \times 3^7$?


  $3^{11}$
• What is the result of $5^8 \div 5^3$?


  $5^5$
• What is the result of $3^5 \text{ }^2$?


  $3^{10}$
• What is the result of $5^2 \times 5^4$?


  $5^6$
• What is the result of $4^3 \div 4^1$?


  $4^2$
• What is the result of $2^8 \text{ }^3$?


  $2^{24}$
• What is the result of $3^2 \times 3^5 \div 3^3$?


  $3^4$
• What is the value of $5^7 \times 5^{-3} \div 5^2$?


  Calculate the value instead of writing it as a power
• What is the process for simplifying $5^7 \times 5^{-3} \div 5^2$?


  1. Combine the powers: $5^{7 - 3 - 2}$
  2. Simplify to get $5^{2}$.
  3. Calculate the value: $5^2 =$$25$.
• What is the result of subtracting the powers when dividing \(7^7\) by \(3^3\)?

  It gives \(3^4\)
• What is the value of \(5^7 \times 5^{-3}\) divided by \(5^2\)?

  25
• How do you simplify \(5^7 \times 5^{-3}\)?

  You add the exponents to get \(5^4\)
• What is the result of \(5^4\) divided by \(5^2\)?


  It simplifies to \(5^2\)
• What is the value of \(5^2\)?


  25
• What happens when you have a negative power in a fraction?

  • You can treat it like a normal number.
  • For example, \(5^{-3}\) in the denominator becomes \(5^3\) in the numerator.
• How do you handle a negative exponent when dividing powers?

  You add the absolute value of the negative exponent to the exponent in the numerator.
• What are the steps to simplify \(3^4 \times 3^3\) divided by \(3^5\)?

  1. Add the exponents: \(3^7\)
  2. Subtract the exponent in the denominator: \(3^{7-5} = 3^2\)
  3. The value of \(3^2\) is 9.
• What is the value of \(3^2\)?

  9
• What does \(a^0\) equal for any non-zero number \(a\)?

  It equals 1.
• Why does any number divided by itself equal 1?

  Because it represents the same quantity, thus simplifying to 1.
• What is the rule for any number raised to the power of zero?

  • Any non-zero number raised to the power of zero equals 1.
  • This applies universally, e.g., \(5^0 = 1\), \(49^0 = 1\).
• What is the significance of \(3^0\)?

  It equals 1.
• What should you be careful about when interpreting \(0\) as a power?

  It is not a degree symbol; it indicates a power of zero.