Exponents Base same

 

Concepts

Exponents, Power of a Power Rule $(a^m{)}^n=a^{m\times n}$, Algebraic Manipulation

Explanation

To solve the equation $7^{98}=x^x$, we need to manipulate the left side of the equation so that it is in the form $a^a$. Once both sides have the same base and exponent (where the base equals the exponent), we can identify the value of $x$ by comparison.

Step-By-Step Solution

Step 1

Identify the given equation:

$$7^{98}=x^x$$

Step 2

Look at the exponent $98$. We want to rewrite $98$ as a product of two numbers such that one of those numbers, when used as a power for $7$, results in a value equal to the remaining part of the exponent.
We know that:

$$98=2\times 49$$

Step 3

Substitute $2\times 49$ for the exponent in the original equation using the power of a power rule $(a^{m\times n}=(a^m{)}^n)$:

$$7^{2\times 49}=x^x$$$$(7^2{)}^{49}=x^x$$

Step 4

Calculate the value of $7^2$:

$$7^2=49$$

Step 5

Substitute $49$ back into the equation:

$$49^{49}=x^x$$

Step 6

By comparing both sides of the equation $49^{49}=x^x$, we can see that the base and the exponent on the left are both $49$, which matches the form $x^x$.
Therefore:

$$x=49$$

Final Answer

$x=49$

Aap ka sawal:

3 ki taqat 2 aur 2 ki taqat x, sab mil kar 9 ke barabar hain.
x ki value kya hai?

Isay equation ki form mein likhen:

3² × 2ˣ = 9

Step-by-step:

• 3² = 9

• Equation ban gayi:
  9 × 2ˣ = 9

Ab dono sides ko 9 se divide karein:

2ˣ = 1

Hum jaante hain:

2⁰ = 1

Final Answer:

x = 0

Acha 👍

Ab aap keh rahe ho:

3^(2^x) = 9

Step-by-step:

Hum jaante hain:

9 = 3²

Is liye equation ban gayi:

3^(2^x) = 3²

Base same hai (3), to powers ko equal kar dete hain:

2^x = 2

Ab:

2^x = 2¹

Is liye:

x = 1 ✅

Bohat acha sawal 👍

Chaliye **aasan tareeqay** se samjhtay hain:

Hamare paas step tha:

**2ˣ = 2¹**

### Rule:

Jab **base same ho** (yahan dono ka base **2** hai),

to **powers barabar hoti hain**.

Is liye:

**x = 1**

### Kyun?

* **2¹ = 2**

* Agar **x = 1** rakhen:

  **2ˣ = 2¹ = 2**

Dono sides same ho gayin ✔️

### Final Answer:

**x = 1**

Agar chaho to main exponent ka **simple rule chart** bhi bana deta hoon jo yaad rakhna bohat easy ho.