The quotient when $10^{100}$ is divided by $5^{75}$ is

### Concept & Formula To divide large exponential numbers with different bases, break down the composite bases into their prime factors. This allows you to group terms with identical prime bases and clearly apply the quotient rule for exponents. $$(xy)^n = x^n \cdot y^n$$ $$\frac{a^m}{a^n} = a^{m-n}$$ ### Step-by-Step Solution * **Given:** Expression to evaluate: $$\frac{10^{100}}{5^{75}}$$ * **Calculation / Deduction:** 1. The base of the numerator, 10, is a composite number. Break it down into its prime factors: $10 = 2 \times 5$. 2. Substitute this back into the numerator: $$\frac{(2 \times 5)^{100}}{5^{75}}$$ 3. Apply the power of a product rule to distribute the exponent: $$\frac{2^{100} \times 5^{100}}{5^{75}}$$ 4. Now, group the terms with base 5 and apply the quotient rule of exponents: $$2^{100} \times 5^{100 - 75}$$ $$2^{100} \times 5^{25}$$ 5. This is the correct mathematical quotient in its simplest prime form, but we must manipulate it to match one of the given options. Let's factor the $2^{100}$ to pair it specifically with the $5^{25}$ term. $$2^{100} = 2^{75} \times 2^{25}$$ 6. Substitute this into our expression: $$2^{75} \times 2^{25} \times 5^{25}$$ 7. Group the terms sharing the common exponent of 25: $$2^{75} \times (2 \times 5)^{25}$$ $$2^{75} \times 10^{25}$$ This matches option (d). ### Exam Strategy & Shortcut **Reverse Engineering Options:** After reaching the simplified prime form $2^{100} \times 5^{25}$, quickly glance at the options rather than guessing how to mathematically reconstruct the composite base. Test Option (a): $2^{25} \times 10^{75} = 2^{25} \times (2 \times 5)^{75} = 2^{100} \times 5^{75}$ (Incorrect). Test Option (d): $2^{75} \times 10^{25} = 2^{75} \times (2 \times 5)^{25} = 2^{100} \times 5^{25}$ (Correct!). Testing the options directly against your simplified prime-factored result is generally faster and less error-prone than intuitively rebuilding the expression. ### Common Pitfall A frequent mistake is attempting to blindly divide the exponents directly (e.g., subtracting 75 from 100 to get an exponent of 25) without aligning the bases first. This leads students to incorrectly choose option (b) $10^{25}$, completely ignoring the base 2 component that was separated during division. ### Final Answer **Therefore, the correct answer is $2^{75} \times 10^{25}$.**