Question

which of the following is equal to $5^{1/3}$? a. $sqrt{5cdot3}$ b. $5^3$ c. $sqrt3{5}$ d. $sqrt{5}$

Explanation:

Step1: Recall the exponent - root relationship

The general formula for converting a fractional exponent to a root is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\). When \(m = 1\), this formula simplifies to \(a^{\frac{1}{n}}=\sqrt[n]{a}\).

Step2: Apply the formula to \(5^{\frac{1}{3}}\)

In the expression \(5^{\frac{1}{3}}\), we have \(a = 5\), \(m=1\) and \(n = 3\). Using the formula \(a^{\frac{1}{n}}=\sqrt[n]{a}\), we substitute the values: \(5^{\frac{1}{3}}=\sqrt[3]{5}\).

Now let's analyze each option:

• Option A: \(\sqrt{5\cdot3}=\sqrt{15}\), which is not equal to \(5^{\frac{1}{3}}\).
• Option B: \(5^{3}=125\), which is not equal to \(5^{\frac{1}{3}}\).
• Option C: \(\sqrt[3]{5}\) is equal to \(5^{\frac{1}{3}}\) as we derived above.
• Option D: \(\sqrt{5}=5^{\frac{1}{2}}\), which is not equal to \(5^{\frac{1}{3}}\).

Answer:

C. \(\sqrt[3]{5}\)