If 3^x \cdot 9^2 = 3^{10}, what is the value of x? — GMAT

If $3^x \cdot 9^2 = 3^{10}$ , what is the value of $x$ ?

Answer: 6

The value of

x

is 6.

Rewriting

9^2

(3^2)^2 = 3^4

converts the expression to a common base of

3

. Applying the product rule gives

3^x \cdot 3^4 = 3^{x+4}

. Setting the exponent

x+4

equal to

10

yields

x = 6

Express all powers using the common base 3

9^2 = (3^2)^2 = 3^4

Converting terms to a common base allows application of exponent rules.

Apply the product rule for exponents

3^x \cdot 3^4 = 3^{x+4}

When multiplying terms with identical bases, add the exponents.

Equate the exponents and solve for x

x + 4 = 10 \implies x = 6

b^m = b^n

for

b > 0

and

b \neq 1

, then

m = n

Exponent rules with base conversion and product of powers

Question