Final answer:
To evaluate the expression 03 (81) without a calculator, we need to find the cube root of 81, which is approximately 4.326. To evaluate the expression log(0.1) without a calculator, we need to find the logarithm of 0.1, which is -1.
Step-by-step explanation:
a) Expression: 03 (81)
To evaluate this expression without a calculator, we need to find the cube root of 81. The cube root of a number is the value that, when multiplied by itself three times, gives the original number. In this case, we want to find the number that, when multiplied by itself three times, equals 81.
Observation: 4 × 4 × 4 = 64, and 5 × 5 × 5 = 125. Therefore, the cube root of 81 must lie between 4 and 5.
Question: What number between 4 and 5, when multiplied by itself three times, gives us 81?
Response: By guessing and checking, we find that 4.326 multiplied by itself three times is approximately equal to 81. Therefore, the cube root of 81 is approximately 4.326.
b) Expression: log(0.1)
To evaluate this expression without a calculator, we need to find the logarithm of 0.1. The logarithm of a number is the power to which another fixed value (called the base) must be raised to produce that number. In this case, we want to find the power to which the base 10 must be raised to give us 0.1.
Observation: The logarithm of a number between 0 and 1 is negative because the base must be raised to a negative power to produce the number.
Question: What power to which the base 10 must be raised to give us 0.1?
Response: By applying the properties of logarithms, we can rewrite the expression as log(0.1) = log(1/10) = log(1) - log(10) = 0 - 1 = -1. Therefore, the logarithm of 0.1 is -1.
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