Answer

Verified

440.7k+ views

**Hint**: We have to compare both the terms to find which one is bigger. To do so, we will write them in the simplest form, that is, \[2.5=\dfrac{5}{2}\] and \[1.25=\dfrac{5}{4}.\] Then we can compare how \[{{\left( 2.5 \right)}^{6}}\] and \[{{\left( 1.25 \right)}^{12}}\] are related to each other by finding the value of \[{{\left( 2.5 \right)}^{6}}=\dfrac{{{5}^{6}}}{{{2}^{6}}}\] and \[{{\left( 1.25 \right)}^{12}}=\dfrac{{{5}^{12}}}{{{2}^{24}}}.\]

**:**

__Complete step-by-step answer__We are asked that which one is larger out of \[{{\left( 2.5 \right)}^{6}}\] and \[{{\left( 1.25 \right)}^{12}}.\] First, we will simplify both the terms.

\[2.5=\dfrac{25}{10}\]

On solving, we get,

\[\Rightarrow 2.5=\dfrac{5}{2}\]

So,

\[{{\left( 2.5 \right)}^{6}}={{\left( \dfrac{5}{2} \right)}^{6}}\]

We know that,

\[{{\left( \dfrac{a}{b} \right)}^{k}}=\dfrac{{{a}^{k}}}{{{b}^{k}}}\]

So, we can write,

\[\Rightarrow {{\left( 2.5 \right)}^{6}}=\dfrac{{{5}^{6}}}{{{2}^{6}}}.....\left( ii \right)\]

Similarly, we can write 1.25 as,

\[1.25=\dfrac{120}{10}\]

On solving, we will get,

\[\Rightarrow 1.25=\dfrac{5}{4}\]

So,

\[\Rightarrow {{\left( 1.25 \right)}^{12}}={{\left( \dfrac{5}{4} \right)}^{12}}\]

We know that,

\[{{\left( \dfrac{a}{b} \right)}^{k}}=\dfrac{{{a}^{k}}}{{{b}^{k}}}\]

\[\Rightarrow {{\left( 1.25 \right)}^{12}}=\dfrac{{{5}^{12}}}{{{4}^{12}}}\]

As we know that, \[{{2}^{2}}=4,\] we can write,

\[\Rightarrow {{\left( 1.25 \right)}^{12}}=\dfrac{{{5}^{12}}}{{{\left( {{2}^{2}} \right)}^{12}}}\]

We know that, \[{{\left( {{a}^{b}} \right)}^{c}}={{a}^{bc}}\]

\[\Rightarrow {{\left( 1.25 \right)}^{12}}=\dfrac{{{5}^{12}}}{{{2}^{24}}}\]

Now,

\[{{\left( 1.25 \right)}^{12}}=\dfrac{{{5}^{6+6}}}{{{2}^{6+18}}}\]

As, \[{{a}^{x+y}}={{a}^{x}}.{{a}^{y}},\] so

\[\Rightarrow {{\left( 1.25 \right)}^{12}}=\dfrac{{{5}^{6}}}{{{2}^{6}}}\times \dfrac{{{5}^{6}}}{{{2}^{18}}}......\left( ii \right)\]

Now, we know that,

\[{{5}^{2}}=25\]

\[{{2}^{5}}=2\times 2\times 2\times 2\times 2=32\]

Clearly, 25 < 32

\[\Rightarrow {{5}^{2}}<{{2}^{5}}\]

Simplifying further, we get,

\[\Rightarrow \dfrac{{{5}^{2}}}{{{2}^{5}}}<1\]

Now we know that if a < 1, then \[{{a}^{n}}\] is also less than 1 for all \[n\ge 1.\]

As \[\dfrac{{{5}^{2}}}{{{2}^{5}}}<1,\] so,

\[{{\left( \dfrac{{{5}^{2}}}{{{2}^{5}}} \right)}^{3}}<1\]

Simplifying using \[{{\left( {{a}^{b}} \right)}^{c}}={{a}^{bc}},\] we get,

\[\dfrac{{{5}^{6}}}{{{2}^{18}}}<1.....\left( iii \right)\]

Now, we use (iii) in (ii), we get,

\[\Rightarrow {{\left( 1.25 \right)}^{12}}=\dfrac{{{5}^{6}}}{{{2}^{6}}}\times \dfrac{{{5}^{6}}}{{{2}^{18}}}\]

As, \[\dfrac{{{5}^{6}}}{{{2}^{18}}}<1,\] so,

\[\Rightarrow {{\left( 1.25 \right)}^{12}}<\dfrac{{{5}^{6}}}{{{2}^{6}}}\times 1\]

\[\Rightarrow {{\left( 1.25 \right)}^{12}}<\dfrac{{{5}^{6}}}{{{2}^{6}}}\]

Using (i), we have, \[\dfrac{{{5}^{6}}}{{{2}^{6}}}={{\left( 2.5 \right)}^{6}}\]

So, we get,

\[{{\left( 1.25 \right)}^{12}}<{{\left( 2.5 \right)}^{6}}\]

So, we get, \[{{\left( 2.5 \right)}^{6}}\] is larger than \[{{\left( 1.25 \right)}^{12}}.\]

**Note**: We have to compare \[{{\left( 1.25 \right)}^{12}}\] with \[{{\left( 2.5 \right)}^{6}}.\] So, while expanding \[{{\left( 1.25 \right)}^{12}}=\dfrac{{{5}^{12}}}{{{2}^{24}}}\] into more simple term, we expand the term by keeping in the mind the value of \[{{\left( 2.5 \right)}^{6}}\] which is \[\dfrac{{{5}^{6}}}{{{2}^{6}}}.\] So, we separate this term out and then we will write as \[{{\left( 1.25 \right)}^{12}}=\dfrac{{{5}^{6}}}{{{2}^{6}}}.\dfrac{{{5}^{6}}}{{{5}^{18}}}.\] When we have this, we will check how other term affects the terms.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

10 examples of evaporation in daily life with explanations

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE