If x > 0 satisfies the exponential equation 3^{x+1} + 3^{1-x} = 10 find the value of 8^x +… — JAMB UTME

x > 0

satisfies the exponential equation

3^{x+1} + 3^{1-x} = 10

find the value of

8^x + 4^{x-1}

Answer: 9

Answer

The value of

8^x + 4^{x-1}

is 9.

Applying index laws to

3^{x+1} + 3^{1-x} = 10

yields

3(3^x) + \frac{3}{3^x} = 10

. Substituting

u = 3^x

gives

3u^2 - 10u + 3 = 0

, which factors to

(3u - 1)(u - 3) = 0

, yielding

u = 3

u = \frac{1}{3}

. Thus

x = 1

x = -1

. Given

x > 0

x = 1

. Substituting

x = 1

into

8^x + 4^{x-1}

gives

8^1 + 4^0 = 8 + 1 = 9

Step-by-Step Solution

Apply the product and negative power laws of indices to separate the terms in the given equation.

3^{x+1} = 3^1 \cdot 3^x = 3(3^x)

and

3^{1-x} = 3^1 \cdot 3^{-x} = \frac{3}{3^x}

, making the equation

3(3^x) + \frac{3}{3^x} = 10

According to the laws of indices,

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

and

a^{-n} = \frac{1}{a^n}

Substitute

u = 3^x

into the equation and clear the fraction to form a standard quadratic equation.

3u + \frac{3}{u} = 10 \implies 3u^2 - 10u + 3 = 0

Multiplying through by

u

eliminates the fraction and forms a quadratic in terms of

u

Factor the quadratic equation

3u^2 - 10u + 3 = 0

to find the values of

u

(3u - 1)(u - 3) = 0 \implies u = 3

u = \frac{1}{3}

Factoring by splitting the middle term gives the linear factors.

Equate

3^x

to the values of

u

and apply the constraint

x > 0

3^x = 3^1 \implies x = 1

and

3^x = 3^{-1} \implies x = -1

. Selecting the positive root gives

x = 1

Equating exponents with matching base 3 gives the solutions for

x

Substitute

x = 1

into the target expression

8^x + 4^{x-1}

and simplify.

8^1 + 4^{1-1} = 8 + 4^0 = 8 + 1 = 9

By the zero index law, any non-zero base raised to the power 0 equals 1 (

a^0 = 1

Key Concept

Solving quadratic-form exponential equations using index laws and applying the zero index rule.

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