Question

assessment practice

1. rewrite each radical using rational exponents.

a. (sqrt6{z^5})
b. (sqrt{5})
c. (sqrt3{2^4})
d. (sqrt4{2})

Explanation:

Step1: Recall the rule for converting radicals to rational exponents

The formula for converting a radical \(\sqrt[n]{a^m}\) to a rational exponent is \(a^{\frac{m}{n}}\), where \(n\) is the index of the radical and \(m\) is the exponent of the radicand. If there is no visible exponent on the radicand, it is assumed to be \(1\), and if there is no visible index on the radical (like a square root), the index is assumed to be \(2\).

Step2: Solve part (a) \(\sqrt[6]{z^{5}}\)

Using the formula \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), here \(n = 6\) and \(m=5\), so \(\sqrt[6]{z^{5}}=z^{\frac{5}{6}}\)

Step3: Solve part (b) \(\sqrt{5}\)

For a square root, \(n = 2\) and the exponent of the radicand \(5\) is \(1\) (since \(5 = 5^1\)). Using the formula \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), we get \(\sqrt{5}=\sqrt[2]{5^1}=5^{\frac{1}{2}}\)

Step4: Solve part (c) \(\sqrt[3]{2^{4}}\)

Using the formula \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), here \(n = 3\) and \(m = 4\), so \(\sqrt[3]{2^{4}}=2^{\frac{4}{3}}\)

Step5: Solve part (d) \(\sqrt[4]{2}\)

For the radicand \(2\), the exponent is \(1\) (since \(2=2^1\)) and the index \(n = 4\). Using the formula \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), we get \(\sqrt[4]{2}=\sqrt[4]{2^1}=2^{\frac{1}{4}}\)

Answer:

a. \(z^{\frac{5}{6}}\)
b. \(5^{\frac{1}{2}}\)
c. \(2^{\frac{4}{3}}\)
d. \(2^{\frac{1}{4}}\)