Solve for the following equation $x$: $3({{2}^{x}}+1)-{{2}^{x+2}}+5=0$
Last updated date: 17th Mar 2023
•
Total views: 305.1k
•
Views today: 8.85k
Answer
305.1k+ views
Hint: Take the equation $3({{2}^{x}}+1)-{{2}^{x+2}}+5=0$. Simplify the equation. Also, apply the product rule. You will get the answer.
Complete step-by-step answer:
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.
$\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
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.
In the question we have to find the value of $x$.
We have been given the equation,
$3({{2}^{x}}+1)-{{2}^{x+2}}+5=0$
$3\times {{2}^{x}}+3-{{2}^{x+2}}+5=0$
Simplifying in a simple manner we get,
$3\times {{2}^{x}}-{{2}^{x+2}}+8=0$
$\begin{align}
&3\times {{2}^{x}}-{{2}^{x}}{{2}^{2}}=-8 \\
&3\times {{2}^{x}}-{{2}^{x}}4=-8 \\
\end{align}$ ……………… (Using product law)
$\begin{align}
&3\times {{2}^{x}}-{{2}^{x}}4=-8 \\
&-{{2}^{x}}=-8 \\
\end{align}$
${{2}^{x}}=8$
We know that, ${{2}^{3}}=8$.
So substituting in above equation we get,
${{2}^{x}}={{2}^{3}}$
$x=3$
Therefore, we get the value of $x$ is $3$.
Note: Read the question carefully. Don’t confuse yourself. You should be clear with the concept of law of exponents. Also, while simplifying do not miss any term. Kindly avoid the mistakes of signs. Solve the sum step by step so that nothing will be missed.
Complete step-by-step answer:
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.
$\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
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.
In the question we have to find the value of $x$.
We have been given the equation,
$3({{2}^{x}}+1)-{{2}^{x+2}}+5=0$
$3\times {{2}^{x}}+3-{{2}^{x+2}}+5=0$
Simplifying in a simple manner we get,
$3\times {{2}^{x}}-{{2}^{x+2}}+8=0$
$\begin{align}
&3\times {{2}^{x}}-{{2}^{x}}{{2}^{2}}=-8 \\
&3\times {{2}^{x}}-{{2}^{x}}4=-8 \\
\end{align}$ ……………… (Using product law)
$\begin{align}
&3\times {{2}^{x}}-{{2}^{x}}4=-8 \\
&-{{2}^{x}}=-8 \\
\end{align}$
${{2}^{x}}=8$
We know that, ${{2}^{3}}=8$.
So substituting in above equation we get,
${{2}^{x}}={{2}^{3}}$
$x=3$
Therefore, we get the value of $x$ is $3$.
Note: Read the question carefully. Don’t confuse yourself. You should be clear with the concept of law of exponents. Also, while simplifying do not miss any term. Kindly avoid the mistakes of signs. Solve the sum step by step so that nothing will be missed.
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths 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

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE
