Question

working with radicals
complete the table below. each expression should be written in radical notation, written with rational exponents and evaluated using the calculator. the first one is done for you.

written in radical notation

written using rational exponents

evaluated to two decimal places

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$sqrt{2}$

$2^{\frac{1}{2}}$

1.41

$sqrt8{7^7}$

$31^{\frac{5}{4}}$

$sqrt7{15^6}$

$19^{\frac{9}{6}}$

$21^{\frac{8}{9}}$

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Explanation:

Step1: Fix 2nd row rational exponent

The radical $\sqrt[8]{7^7}$ converts to $7^{\frac{7}{8}}$.

Step2: Evaluate 2nd row value

Calculate $7^{\frac{7}{8}} = e^{\frac{7}{8}\ln(7)} \approx e^{\frac{7}{8} \times 1.9459} \approx e^{1.7027} \approx 5.50$

Step3: 3rd row rational exponent

The radical $\sqrt[7]{15^6}$ converts to $15^{\frac{6}{7}}$.

Step4: Evaluate 3rd row value

Calculate $15^{\frac{6}{7}} = e^{\frac{6}{7}\ln(15)} \approx e^{\frac{6}{7} \times 2.7080} \approx e^{2.3211} \approx 10.20$

Step5: 4th row radical notation

The rational exponent $19^{\frac{9}{6}}$ simplifies to $19^{\frac{3}{2}} = \sqrt{19^3}$ (or $\sqrt[6]{19^9}$).

Step6: Evaluate 4th row value

Calculate $19^{\frac{3}{2}} = 19 \times \sqrt{19} \approx 19 \times 4.3589 \approx 82.82$

Step7: 5th row radical notation

The rational exponent $21^{\frac{8}{9}}$ converts to $\sqrt[9]{21^8}$.

Step8: Evaluate 5th row value

Calculate $21^{\frac{8}{9}} = e^{\frac{8}{9}\ln(21)} \approx e^{\frac{8}{9} \times 3.0445} \approx e^{2.7062} \approx 15.06$

Answer:

Written in radical notation	Written using rational exponents	Evaluated to two decimal places
$\sqrt[8]{7^7}$	$7^{\frac{7}{8}}$	5.50
$\sqrt[7]{15^6}$	$15^{\frac{6}{7}}$	10.20
$\sqrt{19^3}$ (or $\sqrt[6]{19^9}$)	$19^{\frac{9}{6}}$ (or $19^{\frac{3}{2}}$)	82.82
$\sqrt[9]{21^8}$	$21^{\frac{8}{9}}$	15.06