Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Negative Base. $$(-5.33)^{3}$$

Short answer

\( (-5.33)^{3} = -151 \) (To three significant digits.)

Step-by-step solution

Understanding Exponentiation of a Negative Number

When raising a negative number to an odd power, the result will be negative. This is because multiplying a negative number an odd number of times will always result in a negative product. In the case of \( (-5.33)^{3} \), since the exponent is 3, which is odd, the final answer will also be negative.

Calculating the Cubic Power

To calculate \( (-5.33)^{3} \) means to multiply \( -5.33 \) by itself three times. Compute this using a calculator or manually: \(-5.33 \times -5.33 \times -5.33\).

Consider Significant Figures

Since the base number \( -5.33 \) has three significant digits, the final answer should also be rounded to three significant digits if necessary.

Final Computation

Perform the multiplication to get \( (-5.33)^{3} = -5.33 \times -5.33 \times -5.33 = -151.393 \). The answer, to three significant digits, is \(-151\).

Key concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. The exponent tells us how many times to multiply the base by itself. For example, in the expression \( 2^4 \), 2 is the base and 4 is the exponent, meaning that 2 needs to be multiplied by itself 4 times: \( 2 \times 2 \times 2 \times 2 = 16 \).

When working with exponentiation, it is important to understand that different rules apply depending on the nature of the base (positive or negative) and whether the exponent is even or odd, as these factors will influence the sign and magnitude of the result.

Negative Base Exponents

Negative base exponents bring an additional level of complexity to exponentiation. When the base is negative, the outcome's sign depends on whether the exponent is odd or even. With an odd exponent, negative bases will result in a negative product - as seen with \( (-5.33)^3 \), while an even exponent will render a positive result due to the negative signs canceling each other out.

For instance, \( (-2)^3 = -8 \) because an odd number (3) of negative terms are multiplied, resulting in a negative. However, \( (-2)^4 = 16 \) because the negative base is multiplied an even number (4) of times, resulting in a positive outcome.

Significant Figures in Mathematics

Significant figures in mathematics refer to the digits in a number that carry meaning contributing to its precision. They include all non-zero digits, zeroes between non-zero digits, and any trailing zeroes in the decimal portion. Leading zeroes are not considered significant.

In the context of calculations, significant figures become important when managing the precision of results. When multiplying numbers like in the expression \( (-5.33)^3 \), the number of significant digits in the answer is determined by the original number with the least number of significant figures—in this case, 5.33 with three significant digits. Therefore, the end product of the exponentiation should also be expressed with three significant figures to maintain the proper level of precision, which is why we round \( -151.393 \) to \( -151 \).

Cubic Power

Cubic power specifically refers to raising a base number to the third power, which is indicated by an exponent of 3. To calculate the cubic power of a number, you simply multiply the base by itself twice. Computationally, if we have a base of \( a \), then \( a^3 = a \times a \times a \).

This operation is particularly important in geometry and physics, where it is often used to compute the volume of cubes and other 3D shapes. For example, if each edge of a cube measures \( a \) units in length, then the volume (V) is \( V = a^3 \). And as with other forms of exponentiation, when dealing with significant digits in calculations involving cubic power, ensure that your final result reflects the appropriate level of precision as dictated by the rules of significant figures.