If 25^a = 125^b, where a and b are positive constants, what is the value of \frac{a}{b}? — SAT

If $25^a = 125^b$ , where $a$ and $b$ are positive constants, what is the value of $\frac{a}{b}$ ?

Answer: 1.5

1.5

To find the value of

\frac{a}{b}

, we can rewrite the bases of the equation

25^a = 125^b

using a common base of 5:

(5^2)^a = (5^3)^b

. Applying the exponent power rule gives

5^{2a} = 5^{3b}

. Since the bases are identical, their exponents must be equal, so

2a = 3b

. Dividing both sides of this equation by

2b

yields the ratio

\frac{a}{b} = \frac{3}{2}

, which is

1.5

Express the bases as powers of 5

(5^2)^a = (5^3)^b

To solve the equation, express both sides with a common base of 5 since

25 = 5^2

and

125 = 5^3

Apply the power rule of exponents

5^{2a} = 5^{3b}

According to the exponent rules,

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

Set the exponents equal to each other

2a = 3b

Because the bases on both sides of the equation are equal and positive, their exponents must also be equal.

Solve for the ratio

\frac{a}{b} = 1.5

Divide both sides of the equation

2a = 3b

2b

to isolate the ratio

\frac{a}{b}

Solving exponential equations by expressing bases in terms of a common base and equating exponents.

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