Problem 32 - Technical Mathematics with Calculus

Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Negative Base. $$(-5.84)^{2}$$

Step-by-Step Solution

Answer

34.1

Step 1: Understand the exponent rules

When you raise a negative base to an even power, the result is positive. Apply the exponent without changing the sign of the base first.

Step 2: Calculate the square of 5.84

Multiply 5.84 by itself to get the square.

Step 3: Round to significant digits

Ensure the answer retains the same number of significant digits as the original number. Since 5.84 has three significant digits, the result should also have three significant digits.

Step 4: Attach the correct sign

Since the base was negative and power is even, attach a positive sign to the result.

Key Concepts

Negative Bases and Exponents

Understanding the behavior of exponents when combined with negative bases is crucial in mathematics. When a negative base is raised to an even exponent, the resulting value is positive because multiplying an even number of negative factors will always produce a positive product. Conversely, a negative base raised to an odd exponent results in a negative product, because an odd number of negative factors yield a negative result.

For instance, if we have the expression \( (-5.84)^2 \), we are raising the negative base -5.84 to an even power of 2. The calculation involves two steps: ignoring the sign of the base and multiplying the absolute value by itself. In this case, \( 5.84 \times 5.84 \). Though the base is negative, the square of it becomes positive. However, if our exponent were odd, the result would retain the negative sign.

Exponent Rules

Grasping exponent rules is vital for correctly interpreting and simplifying mathematical expressions. One such rule, as seen in the exercise with negative bases, is that an even exponent nullifies the negative sign of a base. Other key rules include the product rule (\( a^m \times a^n = a^{m+n} \) where \( a \) is a base and \( m \) and \( n \) are exponents), the quotient rule (\( \frac{a^m}{a^n} = a^{m-n} \)), and the power of a power rule (\( (a^m)^n = a^{m \times n} \) ).

Moreover, understanding that any number with an exponent of zero is one (\( a^0 = 1 \) if \( a eq 0 \) ) and that a negative exponent indicates a reciprocal (\( a^{-n} = \frac{1}{a^n} \) if \( a eq 0 \) and \( n > 0 \) ) can simplify complex calculations. Keep in mind that when applying these rules, operations must be performed step by step while adhering to the order of operations.

Rounding Significant Digits

Rounding to significant digits is crucial for accurately representing the precision of a number in scientific and mathematical contexts. The number of significant digits in a figure indicates its precision; more significant digits imply greater precision. When you perform arithmetic operations, it is important to round the results to maintain the correct number of significant digits.

In the given problem, \( (-5.84)^2 \), we square 5.84, but must ensure the result is reported with the same level of precision as the original number, which has three significant digits. After multiplying, we would round our answer to retain this precision. This often requires us to round up or down based on the value of the digit following our last significant digit. If this next digit is five or greater, we round up, and if it's four or less, we round down. This approach ensures that the numbers we use in mathematical expressions and scientific measurements remain as accurate and meaningful as possible.