# Can you Solve this? | Math Olympiad

TLDRIn this Math Olympiad video, the presenter tackles an exponential problem where 256 to the power of n equals 1/n. The solution involves simplifying the equation by taking the nth root of both sides, leading to 4 to the power of 4 equals 1/n to the power of 1/n. By factoring 256 into 4 to the power of 4 and applying the property that x to the power of x equals n to the power of n for n greater than or equal to 1, the presenter deduces that n equals 1/4. The solution is verified by substituting n back into the original equation, confirming that n = 1/4 is indeed the correct answer.

### Takeaways

- ๐งฎ The problem involves solving the equation \(256^n = \frac{1}{n}\) to find the value of \(n\).
- ๐ The initial step is to simplify the equation by taking the nth root of both sides, leading to \(256 = \left(\frac{1}{n}\right)^{\frac{1}{n}}\).
- ๐ Prime factorization of 256 is performed to simplify the equation further, resulting in \(4^4 = \left(\frac{1}{n}\right)^{\frac{1}{n}}\).
- ๐ข By applying the fact that if \(x^x = n^n\) and \(n \geq 1\), then \(x = n\), the equation simplifies to \(4 = \frac{1}{n}\), hence \(n = \frac{1}{4}\).
- ๐ฏ The solution \(n = \frac{1}{4}\) is verified by substituting it back into the original equation to confirm its validity.
- ๐ The video demonstrates the process of prime factorization by breaking down 256 into pairs of twos, resulting in four pairs.
- ๐ The concept of exponents is utilized to equate the powers on both sides of the equation to solve for \(n\).
- ๐ The video provides a step-by-step guide on how to manipulate and simplify exponential equations.
- ๐ The importance of verifying the solution by substituting it back into the original equation is emphasized.
- ๐ The video concludes by confirming that the value of \(n = \frac{1}{4}\) satisfies the given equation, thus solving the problem.

### Q & A

### What is the main mathematical problem discussed in the video?

-The main problem discussed is solving the equation 256^n = 1/n to find the value of n.

### What is the first step taken to solve the equation 256^n = 1/n?

-The first step is to take the nth root on both sides of the equation to simplify it.

### Why are prime factors of 256 important in solving the problem?

-The prime factors of 256 are important because they help in expressing 256 as a power of 4, which simplifies the equation.

### How is 256 expressed in terms of its prime factors?

-256 is expressed as 4^4, which is derived by finding the prime factors and grouping them into pairs.

### What mathematical fact is used to equate the exponents in the problem?

-The fact used is that if x^x = n^n and n โฅ 1, then x = n.

### What is the final value of n obtained in the problem?

-The final value of n obtained is 1/4.

### How is the solution verified in the problem?

-The solution is verified by substituting n = 1/4 back into the original equation and confirming that both sides of the equation are equal.

### What conclusion is drawn after verifying the solution?

-The conclusion is that n = 1/4 is the correct solution, as it satisfies the original equation.

### Why is the exponent 4 cancelled out in the verification step?

-The exponent 4 is cancelled out because when the expression is simplified, both sides of the equation have the same base and exponent.

### What is the final message conveyed by the speaker in the video?

-The final message is to thank the viewers for watching and to encourage them to subscribe for more videos.

### Outlines

### ๐งฎ Solving an Exponential Equation Involving 256 and n

In this paragraph, the problem of solving the equation \(256^n = \frac{1}{n}\) is introduced. The process begins by removing the variable \(n\) from the left-hand side by taking the nth root of both sides. This transforms the equation and allows for further simplification. The exponent laws are applied, reducing the equation to \(256 = \frac{1}{n}\) raised to the power of \(\frac{1}{n}\). The focus then shifts to finding the prime factors of 256, where the method of division by 2 is used iteratively to break down 256 into its prime components. The equation is then rewritten using these factors as \(4^4 = \frac{1}{n}\) raised to the power of \(\frac{1}{n}\), setting the stage for the next steps in solving for \(n\).

### ๐ Applying Exponential Laws to Determine n

This paragraph explains the crucial step of equating the exponents after rewriting the equation. A key mathematical fact is introduced: when \(x^x = n^n\) and \(n \geq 1\), it implies \(x = n\). Using this fact, the equation \(4 = \frac{1}{n}\) is derived, leading to the conclusion that \(n = \frac{1}{4}\). The final step involves verifying this solution by substituting \(n = \frac{1}{4}\) back into the original equation. The verification process shows that the equation holds true, confirming \(n = \frac{1}{4}\) as the correct solution. The paragraph concludes with an invitation to subscribe for more videos.

### Mindmap

### Keywords

### ๐กExponential Problem

### ๐กNth Root

### ๐กPrime Factors

### ๐กExponent Laws

### ๐กEquating Exponents

### ๐กVerification

### ๐กSimplification

### ๐กEquation

### ๐กPower

### ๐กMathematical Proof

### Highlights

Introduction to solving an exponential problem from the Math Olympiad.

The problem statement: 256 to the power of n equals 1/n.

Strategy to isolate variable n by taking nth root on both sides.

Simplification using the properties of exponents.

Finding the prime factors of 256 as a step towards solving the problem.

Methodology to find prime factors by dividing 256 successively by 2.

Result of prime factorization: 256 is expressed as 2^4 * 2^4 * 2^4 * 2^4.

Substitution of 256 with 4 to the power of 4 in the equation.

Application of the fact that x^x = n^n implies x = n for n >= 1.

Derivation of n = 1/4 by equating the exponents.

Verification of the solution by substituting n = 1/4 back into the original equation.

Confirmation that n = 1/4 satisfies the original problem statement.

Conclusion that n = 1/4 is the final solution to the problem.

Encouragement to subscribe for more educational content.

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