Question

Is anybody good at math that can help me..?

Answer 1

Answer:

Option b) 8.∛5

Step-by-step explanation:

Let's break the terms into more simpler values by the help of prime factorisation.

⇒ 8 = 2 × 2 × 2 = 2³

⇒ 320 = 2 × 2 × 2 × 2 × 2 × 2 × 5 = 2⁶ × 5

The power 1/3 refers to cube root.

Substituting in the equation we get:

⇒ ∛ ( 2³ × 2⁶ × 5 )

⇒ ∛ ( 2⁹ × 5 )

⇒ ∛2⁹ × ∛5

⇒ 2³ × ∛5

⇒ 8.∛5

This is the required answer.

Hence Option (b) is the correct answer.

Answer 2

♣ Qᴜᴇꜱᴛɪᴏɴ :

Which of the following is equal to the expression below ?

$\sf{\left(8\cdot 320\right)^{\tfrac{1}{3}}}$

⚪️ 30

⚪️ 8 ∛5

⚪️ 10 ∛5

⚪️ 40

★═════════════════★  

♣ ᴀɴꜱᴡᴇʀ :

⚪️ 30

⚪️ 8 ∛5 ✅

⚪️ 10 ∛5

⚪️ 40

★═════════════════★

♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

$\sf{\left(8\cdot 320\right)^{\tfrac{1}{3}}}$

$\bf{\mathrm{Apply\:exponent\:rule}:\quad \left(a\cdot \:b\right)^n=a^nb^n}$

$\sf{\left(8\cdot \:320\right)^{\tfrac{1}{3}}=8^{\tfrac{1}{3}}\cdot \:320^{\tfrac{1}{3}}}$

$\sf{=8^{\tfrac{1}{3}}\cdot \:320^{\tfrac{1}{3}}}$

$\sf{= \sqrt[3]{8}\:\cdot\:320^{\tfrac{1}{3}}}$

$\sf{= 2\:\cdot\:320^{\tfrac{1}{3}}}$

Prime factorization of 320 :

$\large{ \begin{array}{c|c} \tt 2 & \sf{ 320} \\ \cline{1-2} \tt 2 & \sf { 160} \\ \cline{1-2} \tt 2 & \sf{ 80} \\ \cline{1-2} \tt 2 & \sf{ 40} \\ \cline{1-2} \tt 2 & \sf{ 20 }\\ \cline{1-2} \tt 2 & \sf{ 10 }\\ \cline{1-2} \tt 5 & \sf{ 5} \\ \cline{1-2} & \sf{ 1} \end{array}}$

320 = 2 × 2 × 2 × 2 × 2 × 2 × 5

320 = 2⁶ × 5

$\sf{ 2\:\cdot\:320^{\tfrac{1}{3}}}$

$\sf{=2\left(2^6\cdot \:5\right)^{\tfrac{1}{3}}}$

$\bf{\mathrm{Apply\:exponent\:rule}:\quad \left(a\cdot \:b\right)^n=a^nb^n}$

$\sf{\left(2^6\cdot \:5\right)^{\tfrac{1}{3}}=\left(2^6\right)^{\tfrac{1}{3}}\cdot \:5^{\tfrac{1}{3}}}$

$\sf{=2\cdot \left(2^6\right)^{\tfrac{1}{3}}\cdot \:5^{\tfrac{1}{3}}}$

$\sf{=2\cdot 2^{\tfrac{6}{3}}\cdot \:5^{\tfrac{1}{3}}}$

$\sf{=2\cdot 2^2\cdot \:5^{\tfrac{1}{3}}}$

$\sf{=2\cdot4\cdot \:5^{\tfrac{1}{3}}}$

$\sf{=8\cdot \:5^{\tfrac{1}{3}}}$

We know $\sf{a^{\tfrac{1}{3}}=\sqrt[3]{a}}$

$\sf{8\cdot \:5^{\tfrac{1}{3}}}$

$\sf{=8\cdot \:\:\sqrt[3]{5}}$

Hence $\sf{\left(8\cdot \:320\right)^{\frac{1}{3}}=\sf{8\:\sqrt[3]{5}}}$