If 5^{a + 2} = 25^{b} and a + b = 10, what is the value of b? — SAT

If $5^{a + 2} = 25^{b}$ and $a + b = 10$ , what is the value of $b$ ?

Answer: 4

The correct answer is 4.

The correct answer is 4. By rewriting

25^b

(5^2)^b = 5^{2b}

, we can set the exponents equal to get

a + 2 = 2b

. Substituting

a = 10 - b

from the second equation into this relation yields

(10 - b) + 2 = 2b

. Simplifying this gives

12 = 3b

, which results in

b = 4

Rewrite the exponential equation using a common base.

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

Since

25 = 5^2

, we write

25^b

(5^2)^b = 5^{2b}

to match the base on the left side.

Equate the exponents from the base 5 terms.

a + 2 = 2b

When bases are equal, their exponents must be equal for the equation to hold.

Express

a

in terms of

b

using the linear equation

a + b = 10

a = 10 - b

This allows for substitution into the equation from Step 2 to solve for a single variable.

Substitute

a = 10 - b

into

a + 2 = 2b

and solve for

b

b = 4

Substituting gives

(10 - b) + 2 = 2b \implies 12 - b = 2b \implies 12 = 3b \implies b = 4

Solving systems of equations involving exponential equations by finding a common base.

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