Identify base and exponent: Identify the base and the exponent in the expression -27^(4/3). The base is -27 and the exponent is 4/3.Recognize negative number: Recognize that -27 is a negative number and can be written as (-3)^3. This is because -3 * -3 * -3 equals -27.Apply exponent to base: Apply the exponent to the base by recognizing that when raising a power to another power, you multiply the exponents. So, (-3)^3 raised to the power of 4/3 is equal to (-3)^(3*(4/3)).Simplify exponents: Simplify the exponents by multiplying 3 by 4/3. 3 * (4/3) equals 4. So, (-3)^(3*(4/3)) simplifies to (-3)^4.Calculate final result: Calculate (-3)^4 by multiplying -3 by itself four times. -3 * -3 * -3 * -3 equals 81 because a negative number raised to an even power is positive.

Precalculus

Evaluate rational exponents

Full solution

Q. e)

-27^{\frac{4}{3}}

Identify base and exponent: Identify the base and the exponent in the expression $-27^{4/3}$ . $\newline$ The base is $-27$ and the exponent is $\frac{4}{3}$ .
Recognize negative number: Recognize that $-27$ is a negative number and can be written as $(-3)^3$ . $\newline$ This is because $-3 \times -3 \times -3$ equals $-27$ .
Apply exponent to base: Apply the exponent to the base by recognizing that when raising a power to another power, you multiply the exponents. $\newline$ So, $(-3)^3$ raised to the power of $\frac{4}{3}$ is equal to $(-3)^{(3*\frac{4}{3})}$ .
Simplify exponents: Simplify the exponents by multiplying $3$ by $\frac{4}{3}$ . $3 \times \left(\frac{4}{3}\right)$ equals $4$ .So, $(-3)^{3\times\left(\frac{4}{3}\right)}$ simplifies to $(-3)^4$ .
Calculate final result: Calculate $(-3)^4$ by multiplying $-3$ by itself four times. $\newline$ $-3 \times -3 \times -3 \times -3$ equals $81$ because a negative number raised to an even power is positive.