Finding the square root of 121 means identifying what positive number squared gives 121. The answer is 11, because 11 * 11 = 121.

The product rule tells us to add the exponents when the base is identical. Adding the exponent 1 to the exponent 6 gives a final power of 7, so the answer is 7^7.

To convert the fraction to a decimal, divide 10127 by 100000. This is done by shifting the decimal point five places to the left, which results in 0.10127.

To divide one fraction by another, you must multiply the first fraction by the reciprocal of the second fraction. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$. So, the calculation becomes $\frac{5}{8} \times \frac{4}{3}$. Multiplying the numerators ($5 \times 4 = 20$) and the denominators ($8 \times 3 = 24$) gives $\frac{20}{24}$. This fraction can be simplified by finding the greatest common divisor of 20 and 24, which is 4. Dividing the numerator and denominator by 4 ($20 \div 4 = 5$ and $24 \div 4 = 6$) gives the final simplified answer of $\frac{5}{6}$.

Combining the whole numbers and fractional parts: 555.05 + 55.50 + 5.55 + 5.00 + 0.55 leads mathematically to a total value of 621.65.

The second term has an implicit coefficient of 1. Subtracting 1 from 7 gives 6. The radical part remains the same, producing 6*sqrt(7).

Since the bases are the same, we simply subtract the exponents. Subtracting 3 from 9 leaves an exponent of 6, making the simplified form 4^6.

Simplify both terms first: sqrt(18) = 3*sqrt(2) and sqrt(8) = 2*sqrt(2). Subtracting them gives 3*sqrt(2) - 2*sqrt(2) = 1*sqrt(2), which is just sqrt(2).

The cube root of 343 is a number that, when raised to the third power, yields 343. That number is 7, because 7^3 = 343.

According to the quotient rule for exponents, we subtract the bottom exponent from the top exponent when bases are the same. Subtracting 4 from 9 gives us x^5.