Question

the product³√2·(2)-⅓·¹²√32 equals to​

Answer 1

We can write options a, b, c as,

$\cdots \longrightarrow \text{Option A: \sqrt[12]{32}=(2^5)^\frac{1}{12}=2^{\frac{5}{12}} }$

$\cdots \longrightarrow \text{Option B: \sqrt[12]{2}=2^{\frac{1}{12}} }$

$\cdots \longrightarrow \text{Option C: \sqrt{2}=2^{\frac{1}{2}} }$

First, let's learn about the properties of exponents. These are crucial to solving the problem.

Real exponents

$\cdots \longrightarrow \boxed{(a^{m})^{n}=a^{mn}}$

$\cdots \longrightarrow \boxed{a^{m} \cdot a^{n}=a^{m+n}}$

Rational exponents

$\cdots \longrightarrow \boxed{\text{ {a}^{\frac{m}{n}}=\sqrt[n]{a^{m}} for an integer n \geq 2 }}$

Calculation

$\boxed{\begin{aligned}\sqrt[3]{2} \cdot 2^{-\frac{1}{3}} \cdot \sqrt[12]{32}&=2^{\frac{1}{3}} \cdot 2^{-\frac{1}{3}} \cdot(2^5)^{\frac{1}{12}}\\\\&=2^{\frac{1}{3}} \cdot 2^{-\frac{1}{3}} \cdot 2^{\frac{5}{12}}\\\\&=2^{\frac{1}{3}-\frac{1}{3}+\frac{5}{12}}\\\\&=2^{\frac{5}{12}}.\end{aligned}}$

Hence, option A is correct.

Answer 2

Question :-

Find the product of $\sqrt[3]{2} \times (2)^{ \large - \frac{1}{3}} \times \sqrt[12]{32}$.

Answer :-

• $2 {}^{ \large \frac{5}{12} }$

Step by step explanation :-

If we have to solve the question first we have to know the exponent form.

Required Exponent form,

$\rm\bullet \: {x {}^{a} \times x {}^{b} = x {}^{a + b} }$

$\rm\bullet \: {(x {}^{a})^{b} = x {}^{a \times b} }$

$\rm\bullet \: {x {}^{ \large\frac{a}{b} } = \sqrt[b]{x {}^{a} } }$

Solving the expression,

$\rm:\longmapsto{ \sqrt[3]{2} \times 2 {}^{ \large - \frac{ 1}{3} } \times \sqrt[12]{32} } \: \: \: \: \: \\ \\ \rm:\longmapsto{2 {}^{ \large \frac{1}{3} } \times 2 {}^{ \large - \frac{1}{ 3 } } \times (2 {}^{5} ) {}^{ \large \frac{1}{12} } } \: \: \: \\ \\\rm:\longmapsto{2 {}^{ \large\frac{1}{3} } \times 2 {}^{ \large - \frac{1}{3} } \times 2 {}^{ \large\frac{5}{12} } } \: \: \: \: \: \: \: \: \\ \\ \rm:\longmapsto{2 {}^{ \large\frac{1}{3} - \frac{1}{3} + \frac{5}{12} } } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm:\longmapsto \red{ 2{}^{ \large \frac{5}{12} } } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

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