If 2^x \times 4^3 = 2^9, what is the value of x? — GRE General Test

If $2^x \times 4^3 = 2^9$ , what is the value of $x$ ?

Answer: 3

The value of

x

is 3.

By rewriting

4^3

(2^2)^3 = 2^6

, the equation becomes

2^x \cdot 2^6 = 2^9

. Applying the product rule gives

2^{x+6} = 2^9

, which simplifies to

x + 6 = 9

, yielding

x = 3

Convert all terms to base 2

4^3 = (2^2)^3 = 2^{2 \times 3} = 2^6

To apply exponent rules, expressions should share a common base.

Apply product rule of exponents to the left side of the equation

2^x \times 2^6 = 2^{x+6}

When multiplying exponential terms with the same base, add their exponents:

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

Equate the exponents from both sides

x + 6 = 9

Since the bases are identical and non-zero, the exponents must be equal.

Solve the linear equation for

x

x = 3

Subtract 6 from both sides of the equation.

Combining exponential terms with equal bases using power rules (

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

and

(a^m)^n = a^{mn}

Question