In the equation \frac{25^x}{5^{x-3}} = 125^2, what is the value of x? — SAT

In the equation $\frac{25^x}{5^{x-3}} = 125^2$ , what is the value of $x$ ?

Answer: 3

Answer

The correct answer is 3.

By writing all terms with a base of 5, the equation

\frac{25^x}{5^{x-3}} = 125^2

becomes

\frac{5^{2x}}{5^{x-3}} = 5^6

. Applying the quotient rule of exponents, the left side simplifies to

5^{2x - (x-3)} = 5^{x+3}

. Setting the exponents equal gives

x+3 = 6

, which yields

x = 3

Step-by-Step Solution

Express all terms with a common base of 5.

25^x = (5^2)^x = 5^{2x}

and

125^2 = (5^3)^2 = 5^6

Expressing all exponential terms with the same base allows the exponents to be equated directly once simplified.

Substitute these expressions back into the original equation and simplify the left side using the quotient rule of exponents.

\frac{5^{2x}}{5^{x-3}} = 5^{2x - (x-3)} = 5^{x+3}

. The equation becomes

5^{x+3} = 5^6

The quotient rule states that dividing exponential terms with the same base requires subtracting the exponent of the denominator from the exponent of the numerator:

\frac{b^m}{b^n} = b^{m-n}

Equate the exponents and solve for

x

x + 3 = 6 \implies x = 3

Since the bases on both sides of the equation are equal, their exponents must also be equal: if

b^y = b^z

where

b > 0

and

b \neq 1

, then

y = z

Key Concept

Solving exponential equations by expressing terms with a common base and applying exponent rules.

Estimated Time:1m 30s

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