Question

evaluate the expression.

1. $343^{1/3}$
2. $49^{3/2}$
3. $(sqrt4{81})^3$
4. $sqrt5{-5748}$
5. $sqrt6{23378}$
6. $sqrt3{1080}$

Explanation:

Step1: Rewrite as cube root

$343^{1/3} = \sqrt[3]{343}$

Step2: Find cube root of 343

$\sqrt[3]{343} = 7$ (since $7^3=343$)

Step3: Rewrite as root then power

$49^{3/2} = (\sqrt{49})^3$

Step4: Calculate square root then cube

$\sqrt{49}=7$, $7^3=343$

Step5: Simplify the 4th root first

$\sqrt[4]{81}=3$ (since $3^4=81$)

Step6: Raise result to the 3rd power

$3^3=27$

Step7: Factor out perfect 5th power

$\sqrt[5]{-5748} = \sqrt[5]{-243 \times 23} = \sqrt[5]{(-3)^5 \times 23}$

Step8: Simplify the radical

$\sqrt[5]{(-3)^5 \times 23} = -3\sqrt[5]{23}$

Step9: Factor out perfect 6th power

$\sqrt[6]{23378} = \sqrt[6]{64 \times 365.28125} = \sqrt[6]{2^6 \times 365.28125}$

Step10: Simplify the radical

$\sqrt[6]{2^6 \times 365.28125} = 2\sqrt[6]{365.28125} \approx 2 \times 2.39 = 4.78$ (or keep exact form $2\sqrt[6]{365.28125}$)

Step11: Factor out perfect 7th power

$\sqrt[7]{1080} = \sqrt[7]{128 \times 8.4375} = \sqrt[7]{2^7 \times 8.4375}$

Step12: Simplify the radical

$\sqrt[7]{2^7 \times 8.4375} = 2\sqrt[7]{8.4375} \approx 2 \times 1.36 = 2.72$ (or keep exact form $2\sqrt[7]{8.4375}$)

Answer:

1. $7$
2. $343$
3. $27$
4. $-3\sqrt[5]{23}$ (or approximately $-7.6$)
5. $2\sqrt[6]{365.28125}$ (or approximately $4.78$)
6. $2\sqrt[7]{8.4375}$ (or approximately $2.72$)