Understanding Exponents with Negative Bases

To calculate (-2)^3, you multiply -2 by itself three times: (-2) × (-2) × (-2). First, (-2) × (-2) equals 4, since multiplying two negative numbers results in a positive number. Then, multiplying this result by -2 gives 4 × (-2) = -8. Therefore, the final result of (-2)^3 is -8.

To calculate (-3)^2, you multiply -3 by itself. When a negative number is squared, the result is positive because the two negative signs cancel each other out. Thus, (-3) * (-3) equals 9. Therefore, the value of (-3)^2 is 9.

Any non-zero number raised to the power of zero equals one. This is a fundamental rule of exponents in mathematics. Since -4 is a non-zero number, applying this rule results in (-4)^0 being equal to 1. This principle is consistent across all real numbers, regardless of whether they are positive or negative, thereby confirming that the outcome is always 1 when the exponent is zero.

When evaluating (-5)^1, we are raising -5 to the power of 1. Any number raised to the power of 1 remains the same. Therefore, (-5)^1 equals -5. This means that the value of -5 raised to the first power does not change, resulting in -5 as the outcome.

To find the value of (-2)^4, we raise -2 to the fourth power. This means multiplying -2 by itself four times: (-2) × (-2) × (-2) × (-2). The first two multiplications result in 4, since multiplying two negative numbers gives a positive result: (-2) × (-2) = 4. Then, multiplying the next two: 4 × (-2) = -8 and -8 × (-2) = 16. Therefore, the final result is 16, as raising a negative number to an even power results in a positive value.

To find the value of x^3 when x = -3, we calculate (-3) raised to the power of 3. This means multiplying -3 by itself three times: (-3) × (-3) × (-3). The first two multiplications yield 9, as a negative times a negative equals a positive. Then, multiplying 9 by -3 results in -27. Therefore, the value of x^3 is -27.

Raising -1 to an odd power, such as 5, results in -1. This is because multiplying -1 by itself an odd number of times retains the negative sign. In this case, (-1) × (-1) × (-1) × (-1) × (-1) equals -1, as the first four multiplications yield 1 (since two negatives make a positive), and multiplying that by -1 results in -1. Thus, the final result of (-1)^5 is -1.

To solve the expression (-6)^2 + (-6)^1, first calculate each term separately. (-6)^2 equals 36 because squaring a negative number results in a positive value. Next, (-6)^1 equals -6, as any number to the power of 1 is itself. Adding these results together: 36 + (-6) equals 30. Therefore, the final value of the expression is 30.

To solve the expression (-7)^3 + (-7)^2, we first calculate each term separately. (-7)^3 equals -343, since multiplying -7 by itself three times results in a negative product. Next, (-7)^2 equals 49, as squaring a negative number yields a positive result. Adding these results together, we have -343 + 49, which simplifies to -294. Thus, the final value of the expression is -294.

To find the value of \( y^4 \) when \( y = -2 \), we first raise \(-2\) to the fourth power. Calculating this, we have:

\[
(-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 4 \times 4 = 16
\]

The negative sign is eliminated because raising a negative number to an even power results in a positive number. Therefore, the value of \( y^4 \) is 16.