Answered

Simplify using a single exponent:


  1. $(-5)^2(-5)^3$
  2. $(-2)^{11}(-2)^9(-2)^3$
  3. $6^7 \cdot 6^9 \cdot 6^3$
  4. $\frac{7^4}{7}$
  5. $\frac{10^2}{10^5}$
  6. $\frac{2^4}{2^4}$
  7. $\frac{(-3)^4}{(-3)^5}$
  8. $\frac{(-7)^5}{(-7) \cdot(-7)^4}$
  9. $\frac{8^2}{8^{-2}}$
  10. $\frac{5^4 \cdot 5^9}{5^3 \cdot 5^3}$
  11. $\frac{4^3 \cdot 4^0}{4^2 \cdot 4^3}$
  12. $\frac{9^{12} \cdot 9^3}{9^7 \cdot 9^8}$
  13. $\frac{m^{-2}}{m^{-5}}$
  14. $\frac{3^3}{3^4}$
  15. $\frac{c^{12}}{c^{15}}$
  16. $(-13)^2(-13)^3$
  17. $(-8)^0(-8)^9(-8)^3$
  18. $(-25)^{11}(-25)^3$

Verified Answer :

  • Use the product rule \(b^m \cdot b^n = b^{m+n}\) to combine terms with the same base.
  • Apply the quotient rule \(\frac{b^m}{b^n} = b^{m-n}\) for division.
  • Handle zero and negative exponents using \(b^0 = 1\) and \(b^{-n} = \frac{1}{b^n}\).
  • Simplify all given expressions into the final forms: \(9^{-8}, (-5)^5, (-2)^{23}, 6^{19}, 7^3, 10^{-3}, 2^0, (-3)^{-1}, (-7)^0, 8^4, 5^7, 4^{-2}, 9^0, m^3, 3^{-1}, c^{-3}, (-13)^5, (-8)^{12}, (-25)^{14}\).

Explanation

  1. Understanding the Exponent Laws

We aim to simplify various exponential expressions into a single exponent form. The fundamental rules for exponents we will apply are:

  1. Product of powers with the same base: \(b^m \cdot b^n = b^{m+n}\).
  2. Quotient of powers with the same base: \(\frac{b^m}{b^n} = b^{m-n}\).
  3. Negative exponent: \(b^{-n} = \frac{1}{b^n}\).
  4. Power of zero: \(b^0 = 1\) (where \(b eq 0\)).
  5. Power of one: \(b^1 = b\).

These rules allow us to combine terms systematically. We handle each expression by identifying the base and combining the exponents according to the operations involved.

  1. Simplification of Expressions

Let us simplify the given list of expressions using the identified rules:

  • \(9^{-8}\) is already a single exponent form.
  • \((-5)^2(-5)^3 = (-5)^{2+3} = (-5)^5\)
  • \((-2)^{11}(-2)^9(-2)^3 = (-2)^{11+9+3} = (-2)^{23}\)
  • \(6^7 \cdot 6^9 \cdot 6^3 = 6^{7+9+3} = 6^{19}\)
  • \(\frac{7^4}{7^1} = 7^{4-1} = 7^3\)
  • \(\frac{10^2}{10^5} = 10^{2-5} = 10^{-3}\)
  • \(\frac{2^4}{2^4} = 2^{4-4} = 2^0\)
  • \(\frac{(-3)^4}{(-3)^5} = (-3)^{4-5} = (-3)^{-1}\)
  • \(\frac{(-7)^5}{(-7)^1 \cdot (-7)^4} = \frac{(-7)^5}{(-7)^{1+4}} = \frac{(-7)^5}{(-7)^5} = (-7)^0\)
  • \(\frac{8^2}{8^{-2}} = 8^{2-(-2)} = 8^4\)
  • \(\frac{5^4 \cdot 5^9}{5^3 \cdot 5^3} = \frac{5^{13}}{5^6} = 5^7\)
  • \(\frac{4^3 \cdot 4^0}{4^2 \cdot 4^3} = \frac{4^3}{4^5} = 4^{3-5} = 4^{-2}\)
  • \(\frac{9^{12} \cdot 9^3}{9^7 \cdot 9^8} = \frac{9^{15}}{9^{15}} = 9^0\)
  • \(\frac{m^{-2}}{m^{-5}} = m^{-2-(-5)} = m^3\)
  • \(\frac{3^3}{3^4} = 3^{3-4} = 3^{-1}\)
  • \(\frac{c^{12}}{c^{15}} = c^{12-15} = c^{-3}\)
  • \((-13)^2(-13)^3 = (-13)^{2+3} = (-13)^5\)
  • \((-8)^0(-8)^9(-8)^3 = (-8)^{0+9+3} = (-8)^{12}\)
  • \((-25)^{11}(-25)^3 = (-25)^{11+3} = (-25)^{14}\)
  1. Conclusion

The expressions have been successfully simplified into single exponent forms. The final results are summarized below:

  • \(9^{-8}\)
  • \((-5)^5\)
  • \((-2)^{23}\)
  • \(6^{19}\)
  • \(7^3\)
  • \(10^{-3}\)
  • \(2^0\)
  • \((-3)^{-1}\)
  • \((-7)^0\)
  • \(8^4\)
  • \(5^7\)
  • \(4^{-2}\)
  • \(9^0\)
  • \(m^3\)
  • \(3^{-1}\)
  • \(c^{-3}\)
  • \((-13)^5\)
  • \((-8)^{12}\)
  • \((-25)^{14}\)

Examples

Understanding exponent laws is vital for fields like engineering, computer science, and physics. For example, when measuring extremely large or tiny distances (like those in space or at the atomic level), scientific notation uses exponents to make numbers readable. Similarly, digital data storage capacity—from kilobytes to gigabytes—follows powers of 2. Mastering these rules helps manage, compare, and simplify such quantitative data efficiently.