Question

simplify the following expression: $8^{\frac{5}{3}}$ options: 32, 40, 64, 16

Explanation:

Step1: Rewrite 8 as a power of 2

Since \( 8 = 2^3 \), we can rewrite the expression \( 8^{\frac{5}{3}} \) as \( (2^3)^{\frac{5}{3}} \).

Step2: Apply the power - of - a - power rule

The power - of - a - power rule states that \( (a^m)^n=a^{m\times n} \). For \( (2^3)^{\frac{5}{3}} \), we have \( m = 3 \) and \( n=\frac{5}{3} \), so \( (2^3)^{\frac{5}{3}}=2^{3\times\frac{5}{3}} \).

Step3: Simplify the exponent

Simplify \( 3\times\frac{5}{3} \). The 3 in the numerator and the 3 in the denominator cancel out, leaving us with \( 2^{5} \).

Step4: Calculate \( 2^{5} \)

We know that \( 2^{5}=2\times2\times2\times2\times2 = 32 \).

Answer:

32