What value of n satisfies the equation 2x9yn + 4x2y10 = 8x11y20?

Finding the value of n

To solve the equation 2x⁹yⁿ + 4x²y¹⁰ = 8x¹¹y²⁰, we need to simplify and compare the coefficients of both sides of the equation.

Step 1: Simplify the equation

Firstly, notice that both terms on the left can be factored. Let’s factor out the common term from the left side:

2x⁹yⁿ + 4x²y¹⁰ = 2x²(x⁷yⁿ + 2y¹⁰)

Step 2: Consider the right-hand side

The right side of the equation, 8x¹¹y²⁰, can be rewritten as:

8 = 2 * 4 = 2 * 2² = 2³

So, we can view it as:

8x¹¹y²⁰ = 2³ x¹¹ y²⁰

Step 3: Compare coefficients

From our factorization, let’s equate the simplified left side and the right side:

2x²(x⁷yⁿ + 2y¹⁰) = 2³ x¹¹ y²⁰

This means, dividing both sides by 2:

x²(x⁷yⁿ + 2y¹⁰) = 4xy²⁰

Step 4: Analyze the equations

The resulting expressions imply:

• x² needs to balance with x⁹
• yⁿ and y²⁰ need to balance

Step 5: Determine corresponding powers

For the x terms:

2 + 7 = 11

Confirmed! And for the y terms:

n + 10 = 20

Solving for n gives us:

n = 20 – 10 = 10

Conclusion

The value of n that satisfies the equation

n = 10