Answered

Simplify each expression:


  1. $\frac{(-7)^{5-7}}{(-7) \div(-7)^4}$


  1. $\frac{8^2}{8^4}$


  1. $\frac{4^3 \cdot 4^0}{4^2 \cdot 4^3}$


  1. $\frac{9^{12} \cdot 9^3}{9^7 \cdot 9^8}$


  1. $\frac{3^3}{3^4}$


  1. $\frac{c^{12}}{c^{15}}$


  1. $(-8)^0(-8)^9(-8)^3$


  1. $(-25)^{11}(-25)^3$

Verified Answer :

  • Use the exponent rules: product (\(x^a \cdot x^b = x^{a+b}\)), quotient (\(\frac{x^a}{x^b} = x^{a-b}\)), and zero (\(x^0 = 1\)) to simplify each term.
  • Perform calculations systematically to arrive at the base power for each expression.
  • Results: 14. \(\boxed{-7}\), 15. \(\boxed{\frac{1}{64}}\), 19. \(\boxed{\frac{1}{16}}\), 20. \(\boxed{1}\), 24. \(\boxed{\frac{1}{3}}\), 25. \(\boxed{\frac{1}{c^3}}\), 29. \(\boxed{(-8)^{12}}\), 30. \(\boxed{25^{14}}\).

Explanation

  1. Problem Analysis and Exponent Rules

To simplify expressions with exponents, we rely on fundamental exponent laws:

  1. Product rule: \(x^a \cdot x^b = x^{a+b}\)
  2. Quotient rule: \(\frac{x^a}{x^b} = x^{a-b}\)
  3. Power of a power rule: \((x^a)^b = x^{a \cdot b}\)
  4. Zero exponent rule: \(x^0 = 1\) (for \(x eq 0\))
  5. Negative exponent rule: \(x^{-a} = \frac{1}{x^a}\)

We will apply these systematically to each expression.

  1. Step-by-Step Simplification
  2. \(\frac{(-7)^{5-7}}{(-7) \div (-7)^4} = \frac{(-7)^{-2}}{(-7)^{1-4}} = \frac{(-7)^{-2}}{(-7)^{-3}} = (-7)^{-2 - (-3)} = (-7)^1 = -7\).
  3. \(\frac{8^2}{8^4} = 8^{2-4} = 8^{-2} = \frac{1}{8^2} = \frac{1}{64}\).
  4. \(\frac{4^3 \cdot 4^0}{4^2 \cdot 4^3} = \frac{4^{3+0}}{4^{2+3}} = \frac{4^3}{4^5} = 4^{3-5} = 4^{-2} = \frac{1}{4^2} = \frac{1}{16}\).
  5. \(\frac{9^{12} \cdot 9^3}{9^7 \cdot 9^8} = \frac{9^{12+3}}{9^{7+8}} = \frac{9^{15}}{9^{15}} = 9^{15-15} = 9^0 = 1\).
  6. \(\frac{3^3}{3^4} = 3^{3-4} = 3^{-1} = \frac{1}{3}\).
  7. \(\frac{c^{12}}{c^{15}} = c^{12-15} = c^{-3} = \frac{1}{c^3}\) (for \(c eq 0\)).
  8. \((-8)^0(-8)^9(-8)^3 = 1 \cdot (-8)^{9+3} = (-8)^{12}\).
  9. \((-25)^{11}(-25)^3 = (-25)^{11+3} = (-25)^{14} = 25^{14}\).
  1. Final Summary of Results

The simplified expressions are:

  1. \(-7\)
  2. \(\frac{1}{64}\)
  3. \(\frac{1}{16}\)
  4. \(1\)
  5. \(\frac{1}{3}\)
  6. \(\frac{1}{c^3}\)
  7. \((-8)^{12}\)
  8. \((-25)^{14} = 25^{14}\)

Examples

Exponent rules are fundamental in scientific computing and engineering. For example, when measuring data storage, you often multiply or divide by factors of \(10^3\) (kilobytes, megabytes, gigabytes). Understanding that \(\frac{10^9}{10^6} = 10^3\) allows you to quickly convert between units, ensuring that calculations involving large digital files are handled accurately and efficiently in software development.

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