Given that x and y are real numbers satisfying the simultaneous exponential equations 3^x … — JAMB UTME

Given that

x

and

y

are real numbers satisfying the simultaneous exponential equations

3^x \times 9^y = 81

and

8^x \times 4^y = 256

find the value of

x^2 + y^2

Answer: 5

Answer

The value of

x^2 + y^2

is 5.

Converting all terms to their prime bases (

3

for the first equation and

2

for the second equation) produces the simultaneous linear equations

x + 2y = 4

and

3x + 2y = 8

. Solving these gives

x = 2

and

y = 1

. Substituting these values into

x^2 + y^2

gives

2^2 + 1^2 = 5

Step-by-Step Solution

Convert all terms in the first equation to powers of base 3.

x + 2y = 4

Since

9 = 3^2

and

81 = 3^4

, applying the product law of indices

a^m \times a^n = a^{m+n}

gives

3^{x+2y} = 3^4

. Equating the exponents gives

x + 2y = 4

Convert all terms in the second equation to powers of base 2.

3x + 2y = 8

Since

8 = 2^3

4 = 2^2

, and

256 = 2^8

, applying the laws of indices yields

2^{3x} \times 2^{2y} = 2^8 \implies 2^{3x+2y} = 2^8

. Equating exponents gives

3x + 2y = 8

Solve the system of simultaneous linear equations for

x

and

y

x = 2

and

y = 1

Subtracting

x + 2y = 4

from

3x + 2y = 8

yields

2x = 4 \implies x = 2

. Substituting

x = 2

into

x + 2y = 4

gives

2 + 2y = 4 \implies y = 1

Evaluate the target expression

x^2 + y^2

Substitute

x = 2

and

y = 1

into

x^2 + y^2

to obtain

2^2 + 1^2 = 4 + 1 = 5

Key Concept

Converting exponential terms to common prime bases to reduce exponential equations into linear equations.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic