High School

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Simplify the following expressions:

a) \[ \left(\frac{-12}{8} \sqrt{\frac{6}{5}}\right)\left(\frac{-16}{-27} \sqrt{\frac{35}{14}}\right) \]

b) \[ \left(\frac{24}{30} \sqrt{\frac{55}{15}}\right)\left(\frac{36}{14} \sqrt{\frac{21}{22}}\right) \]

Answer :

Sure! Let's tackle each part of the given question step-by-step to simplify the expressions.

### Part (a)

1. Simplify the fractions:

[tex]\[
\frac{-12}{8} = -\frac{3}{2} \quad \text{(by dividing numerator and denominator by 4)}
\][/tex]

[tex]\[
\frac{-16}{-27} = \frac{16}{27} \quad \text{(signs cancel out)}
\][/tex]

2. Simplify the radicals:

[tex]\[
\sqrt{\frac{6}{5}} \quad \text{and} \quad \sqrt{\frac{35}{14}}
\][/tex]

These are kept as they are because they are already simplified in terms of their form.

3. Combine the fractions and the radicals:

Multiply the simplified fractions:

[tex]\[
-\frac{3}{2} \times \frac{16}{27} = -\frac{48}{54}
\][/tex]

Simplify [tex]\(-\frac{48}{54}\)[/tex] by dividing the numerator and denominator by 6:

[tex]\[
-\frac{8}{9}
\][/tex]

4. Multiply the radicals:

[tex]\[
\sqrt{\frac{6}{5}} \times \sqrt{\frac{35}{14}} = \sqrt{\frac{6}{5} \times \frac{35}{14}}
\][/tex]

Simplify inside the radical:

[tex]\[
\frac{6 \times 35}{5 \times 14} = \frac{210}{70} = 3
\][/tex]

So:

[tex]\[
\sqrt{3}
\][/tex]

5. Final multiplication:

Combine the results from multiplying the simplified fractions and radicals:

[tex]\[
-\frac{8}{9} \times \sqrt{3}
\][/tex]

The numerical value of this expression is approximately [tex]\(-1.54\)[/tex].

### Part (b)

1. Simplify the fractions:

[tex]\[
\frac{24}{30} = \frac{4}{5} \quad \text{(by dividing numerator and denominator by 6)}
\][/tex]

[tex]\[
\frac{36}{14} = \frac{18}{7} \quad \text{(by dividing numerator and denominator by 2)}
\][/tex]

2. Simplify the radicals:

[tex]\[
\sqrt{\frac{55}{15}} \quad \text{and} \quad \sqrt{\frac{21}{22}}
\][/tex]

These radicals remain as is because they are simplified in terms of their form.

3. Combine the fractions and the radicals:

Multiply the simplified fractions:

[tex]\[
\frac{4}{5} \times \frac{18}{7} = \frac{72}{35}
\][/tex]

This fraction doesn't simplify further.

4. Multiply the radicals:

[tex]\[
\sqrt{\frac{55}{15}} \times \sqrt{\frac{21}{22}} = \sqrt{\frac{55 \times 21}{15 \times 22}}
\][/tex]

Simplify inside the radical:

[tex]\[
\frac{1155}{330} = \frac{21}{6} = \frac{7}{2}
\][/tex]

So:

[tex]\[
\sqrt{\frac{7}{2}}
\][/tex]

5. Final multiplication:

Combine the results from multiplying the simplified fractions and radicals:

[tex]\[
\frac{72}{35} \times \sqrt{\frac{7}{2}}
\][/tex]

The numerical value of this expression is approximately [tex]\(3.85\)[/tex].

Thus, the simplified results for the expressions are approximately [tex]\(-1.54\)[/tex] for part (a) and [tex]\(3.85\)[/tex] for part (b).

Thanks for taking the time to read Simplify the following expressions a left frac 12 8 sqrt frac 6 5 right left frac 16 27 sqrt frac 35 14 right b left. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

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