This represents the fundamental product rule of indices. When multiplying terms with the same base 'a', their exponents are added together, resulting in a^(m+n).

Using the product law, 10^3 * 10^2 = 10^(3+2) = 10^5. 10^5 is equal to 1 followed by 5 zeros, which is 100,000.

Applying the product rule of indices, 4^2 * 4^1 = 4^(2+1) = 4^3. Evaluating 4^3 gives 4 * 4 * 4 = 64.

When multiplying terms with the same base, the exponents are added together. Here, the base is x. Adding the exponents 3 and 4 gives x^(3+4) = x^7.

Any non-zero number raised to the power of 0 is 1 (2^0 = 1). Therefore, 2^5 * 1 remains 2^5. Calculating 2^5 gives 2 * 2 * 2 * 2 * 2 = 32.

Using the law of indices (a^m * a^n = a^(m+n)), we add the powers: 4 + (-2) = 2. This gives 5^2. Squaring 5 results in 5 * 5 = 25.

According to the product law of indices, when multiplying expressions with the same base, you add the exponents. Therefore, 3^2 * 3^3 = 3^(2+3) = 3^5.

A negative exponent means taking the reciprocal of the base and making the exponent positive. So, 2^-3 becomes 1/(2^3), which evaluates to 1/8.

The valid interval boundaries are 7/13 (≈ 0.538) and 4/5 (0.80). Checking the values: 1/2 = 0.5 (falls below 0.538), 2/3 ≈ 0.667, 3/4 = 0.75, 5/7 ≈ 0.714. Hence, 1/2 does not belong inside the targeted range.

The largest perfect square that divides 27 is 9. We can write sqrt(27) as sqrt(9 * 3). The square root of 9 is 3, making the simplified form 3*sqrt(3).