Question

prove that

$( \sqrt{3 \times 5 { }^{ - 3} } \div \sqrt[3]{3 {}^{ - 1} } \sqrt{5} ) \times \sqrt[6]{3 \times {5}^{6} } = \frac{3}{5}$
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Answer 1

Answer:

Question:

Prove that:

${ \sf{( \sqrt{3 \times {5}^{ - 3} } \div \sqrt[3]{ {3}^{ - 1} } \sqrt{5}) \times \sqrt[6]{3 \times {5}^{6} } = \frac{3}{5} }}$

Solution:

${ \implies{ \sf{( \sqrt{3 \times {5}^{ - 3} } \div \sqrt[3]{ {3}^{ - 1} } \sqrt{5}) \times \sqrt[6]{3 \times {5}^{6} }}}}$

${ \implies{ \sf{ \bigg( { \big(3 \times {5}^{ - 3}) }^{ \frac{1}{2} } \div { \big( {3}^{ - 1}) }^{ \frac{1}{3} } ({5})^{ \frac{1}{2} } \bigg) \times (5 \times {3}^{ \frac{1}{6} } }}}$

${ \implies{ \sf{ \bigg( {(3)}^{ \frac{1}{2} } \times {(5)}^{ \frac{ - 3}{2} } \div {(3)}^{ \frac{ - 1}{3} } \times {5}^{ \frac{1}{2} } \bigg) \times 5 \times {3}^{ \frac{1}{6} } }}}$

${ \implies{ \sf{ \bigg( ({3})^{ \frac{3 + 2}{6} } \times {(5)}^{ { \frac{ - 4}{2} } } \bigg) \times {3}^{ \frac{1}{6} } \times 5 }}}$

${ \implies{ \sf{ \bigg( {(3)}^{ \frac{5}{6} } \times {5}^{ - 2} \bigg) \times {3}^{ \frac{1}{6} } \times 5}}}$

${ \implies{ \sf{( {3)}^{ \frac{5}{6} + \frac{1}{6} } \times {5}^{ - 1} }}}$

${ \implies{ \sf{ {3}^{ \frac{6}{6} } \times \frac{1}{5} }}}$

${ \implies{ \sf{3 \times \frac{1}{5} }}}$

${ \implies{ \sf{ \frac{3}{5} }}}$

${ \large{ \mathbb{ \blue{HENCE \: \: PROVED ...! }}}}$

Answer 2

Solution!!

$(\sqrt{3\times 5^{-3}}\div \sqrt[3]{3^{-1}}\sqrt{5})\times \sqrt[6]{3\times 5^{6}}=\dfrac{3}{5}$

Take LHS.

$=(\sqrt{3\times 5^{-3}}\div \sqrt[3]{3^{-1}}\sqrt{5})\times \sqrt[6]{3\times 5^{6}}$

Use $\sqrt[n]{a}=\sqrt[mn]{a^{m}}$ to expand the expression.

$=\sqrt[6]{(3\times 5^{-3})^{3}}\div \sqrt[6]{3^{-2}}\times \sqrt[6]{5^{3}})\sqrt[6]{3\times 5^{6}}$

Simplify the radical expression.

$=\sqrt[6]{(3\times 5^{-3})^{3}}\div \sqrt[6]{3^{-2}}\times \sqrt[6]{5^{3}})5\sqrt[6]{3}$

The product of roots with same index is equal to the root of the product.

$=\sqrt[6]{(3\times 5^{-3})^{3}\div 3^{-2}\times 5^{3}}\times 5\sqrt[6]{3}$

In order to raise a product to a power, raise each factor to that power.

$=\sqrt[6]{27\times 5^{-9}\div 3^{-2}\times 5^{3}}\times 5\sqrt[6]{3}$

Dividing by $a^{-n}$ is the same as multiplying by $a^{n}$.

$=\sqrt[6]{27\times 5^{-9}\times 3^{2}\times 5^{3}}\times 5\sqrt[6]{3}$

Write the number in exponential form.

$=\sqrt[6]{3^{3}\times 5^{-9}\times 3^{2}\times 5^{3}}\times 5\sqrt[6]{3}$

Calculate the product using $a^{m}\times a^{n}=a^{m+n}$.

$=\sqrt[6]{3^{5}\times 5^{-6}}\times 5\sqrt[6]{3}$

Simplify the radical expression.

$=5^{-1}\times \sqrt[6]{3^{5}}\times 5\sqrt[6]{3}$

The product of roots with same index is equal to the root of the product.

$=5^{-1}\times \sqrt[6]{3^{5}\times 3}\times 5$

Calculate the product using $a^{m}\times a^{n}=a^{m+n}$.

$=5^{-1}\times \sqrt[6]{3^{6}}\times 5$

Reduce the exponent and index of the radical with 6.

$=5^{-1}\times 3\times 5$

Calculate the product using $a^{m}\times a^{n}=a^{m+n}$.

$=5^{0}\times 3$

Any non-zero expression with the exponent 0 equals 1.

$=1\times 3$

Any expression multiplied by 1 remains the same.

$=3$

3 ≠ 3/5

LHS ≠ RHS