Only Geniuses Can Solve This | A Challenging Exponential Problem

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[tungstwn]

Video title: "Only geniuses can solve this"
Video: Math that any calculus student (and probably algebra 2 student) should be able to solve

[LjSmth]

Question 1: Solve for x in the equation 9^x = 36

Answer: x = 2

Step-by-step solution:

- The equation 9^x = 36 represents an exponential equation, where the base is 9 and the exponent is x.
- To solve for x, we need to find the value of x that makes the equation true.
- We can rewrite 36 as 6^2 and 9 as 3^2.
- Substituting these values into the equation, we get (3^2)^x = 6^2.
- Using the rule of exponents that states (a^m)^n = a^(m*n), we can simplify the equation to 3^(2x) = 6^2.
- To solve for x, we need to make the bases of both sides of the equation the same.
- Since 6 is not a power of 3, we can rewrite 6 as 2*3.
- Substituting this value into the equation, we get 3^(2x) = (2*3)^2.
- Simplifying the right side of the equation, we get 3^(2x) = 2^2 * 3^2.
- Now, both sides of the equation have the same base, 3.
- Using the rule of exponents that states a^m = a^n if and only if m = n, we can equate the exponents of both sides of the equation.
- Therefore, 2x = 2.
- Solving for x, we get x = 1.

Key knowledge:

- Exponents represent repeated multiplication.
- The base of an exponent is the number being multiplied.
- The exponent indicates how many times the base is multiplied by itself.
- The rule of exponents (a^m)^n = a^(m*n) states that when raising a power to another power, the exponents are multiplied.
- The rule of exponents a^m = a^n if and only if m = n states that if two powers with the same base are equal, then their exponents are equal.

Tips:

- To solve exponential equations, try to rewrite the numbers on both sides of the equation with the same base.
- Use the rules of exponents to simplify the equation.
- Equate the exponents of both sides of the equation to solve for the variable.