Question

rational exponents: power of a power rule
simplify.
$left(x^{3}
ight)^{\frac{7}{12}}$
write your answer without parentheses.
assume that the variable represents a positive real num

Explanation:

Step1: Apply power of a power rule

The power of a power rule states that \((a^m)^n = a^{m \times n}\). Here, \(a = x\), \(m = 3\), and \(n=\frac{7}{12}\). So we multiply the exponents: \(3\times\frac{7}{12}\).

Step2: Simplify the exponent

Simplify \(3\times\frac{7}{12}\). We can write \(3\) as \(\frac{3}{1}\), then multiply the numerators and denominators: \(\frac{3\times7}{1\times12}=\frac{21}{12}\). Then simplify \(\frac{21}{12}\) by dividing numerator and denominator by their greatest common divisor, which is 3. So \(\frac{21\div3}{12\div3}=\frac{7}{4}\). So the expression simplifies to \(x^{\frac{7}{4}}\).

Answer:

\(x^{\frac{7}{4}}\)