What is the value of ${{(a+b)}^{2}}$, If ${{5}^{a+b}}=5\times 25\times 125$ ?
A.$25$
B.$28$
C.36
D.$44$
Last updated date: 19th Mar 2023
•
Total views: 304.5k
•
Views today: 7.83k
Answer
304.5k+ views
Hint: We have to find ${{(a+b)}^{2}}$ . For that, we have been given ${{5}^{a+b}}=5\times 25\times 125$ . So put $\log $ on both sides and use the properties of logarithm. After that, you will get the value of $(a+b)$ , then square it and you will get the answer.
Complete step-by-step Solution:
You must have come across the expression ${{3}^{2}}$. Here $3$ is the base and $2$ is the exponent. Exponents are also called Powers or Indices. The exponent of a number tells how many times to use the number in a multiplication. Let us study the laws of the exponent. It is very important to understand how the laws of exponents' laws are formulated.
Product law: According to the product law of exponents when multiplying two numbers that have the same base then we can add the exponents.
${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
where a, m, and n all are natural numbers. Here the base should be the same in both the quantities.
Quotient Law: According to the quotient law of exponents, we can divide two numbers with the same base by subtracting the exponents. In order to divide two exponents that have the same base, subtract the power in the denominator from the power in the numerator.
$\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
Power Law: According to the power law of exponents, if a number raises a power to a power, just multiply the exponents.
${{({{a}^{m}})}^{n}}={{a}^{mn}}$
Exponential form refers to a numeric form that involves exponents. One method to write such a number is by identifying that each position is representing a power (exponent) of $10$. Thus, you can initially break it up into different pieces. Exponents are also called Powers or Indices.
Now we have been given, ${{5}^{a+b}}=5\times 25\times 125$ .
So solving it,
${{5}^{a+b}}=5\times 25\times 125$
Applying $\log $on both sides we get,
$\log {{5}^{a+b}}=\log \left( 5\times 25\times 125 \right)$
We know the properties of $\log $ ,
$\log {{a}^{k}}=k\log a$
$\log (a\times b)=\log a+\log b$
$\log (\dfrac{a}{b})=\log a-\log b$
So now applying above properties we get,
$\begin{align}
& (a+b)\log 5=\log 5+\log 25+\log 125 \\
& (a+b)\log 5=\log 5+\log {{5}^{2}}+\log {{5}^{3}} \\
& (a+b)\log 5=\log 5+2\log 5+3\log 5 \\
\end{align}$
Taking $\log 5$, in RHS we get,
$\begin{align}
& (a+b)\log 5=\log 5(1+2+3) \\
& (a+b)=1+2+3 \\
& (a+b)=6 \\
\end{align}$
Now squaring both sides we get,
$\begin{align}
& {{(a+b)}^{2}}={{6}^{2}} \\
& {{(a+b)}^{2}}=36 \\
\end{align}$
So we get the value of ${{(a+b)}^{2}}$ as $36$ .
The correct answer is option(C).
Note: Read the question carefully. Also, take utmost care that no terms are missing. Do not make silly mistakes while solving. While simplifying, take care that you solve it step by step. Do not confuse while solving.
Complete step-by-step Solution:
You must have come across the expression ${{3}^{2}}$. Here $3$ is the base and $2$ is the exponent. Exponents are also called Powers or Indices. The exponent of a number tells how many times to use the number in a multiplication. Let us study the laws of the exponent. It is very important to understand how the laws of exponents' laws are formulated.
Product law: According to the product law of exponents when multiplying two numbers that have the same base then we can add the exponents.
${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
where a, m, and n all are natural numbers. Here the base should be the same in both the quantities.
Quotient Law: According to the quotient law of exponents, we can divide two numbers with the same base by subtracting the exponents. In order to divide two exponents that have the same base, subtract the power in the denominator from the power in the numerator.
$\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
Power Law: According to the power law of exponents, if a number raises a power to a power, just multiply the exponents.
${{({{a}^{m}})}^{n}}={{a}^{mn}}$
Exponential form refers to a numeric form that involves exponents. One method to write such a number is by identifying that each position is representing a power (exponent) of $10$. Thus, you can initially break it up into different pieces. Exponents are also called Powers or Indices.
Now we have been given, ${{5}^{a+b}}=5\times 25\times 125$ .
So solving it,
${{5}^{a+b}}=5\times 25\times 125$
Applying $\log $on both sides we get,
$\log {{5}^{a+b}}=\log \left( 5\times 25\times 125 \right)$
We know the properties of $\log $ ,
$\log {{a}^{k}}=k\log a$
$\log (a\times b)=\log a+\log b$
$\log (\dfrac{a}{b})=\log a-\log b$
So now applying above properties we get,
$\begin{align}
& (a+b)\log 5=\log 5+\log 25+\log 125 \\
& (a+b)\log 5=\log 5+\log {{5}^{2}}+\log {{5}^{3}} \\
& (a+b)\log 5=\log 5+2\log 5+3\log 5 \\
\end{align}$
Taking $\log 5$, in RHS we get,
$\begin{align}
& (a+b)\log 5=\log 5(1+2+3) \\
& (a+b)=1+2+3 \\
& (a+b)=6 \\
\end{align}$
Now squaring both sides we get,
$\begin{align}
& {{(a+b)}^{2}}={{6}^{2}} \\
& {{(a+b)}^{2}}=36 \\
\end{align}$
So we get the value of ${{(a+b)}^{2}}$ as $36$ .
The correct answer is option(C).
Note: Read the question carefully. Also, take utmost care that no terms are missing. Do not make silly mistakes while solving. While simplifying, take care that you solve it step by step. Do not confuse while solving.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
