If 5^{x+2} - 5^x = 120\sqrt{5}, what is the value of x? — SAT

If $5^{x+2} - 5^x = 120\sqrt{5}$ , what is the value of $x$ ?

Answer: 1.5

1.5 (or 3/2)

By factoring out the common term

5^x

from the left side of the equation, we get

5^x(5^2 - 1) = 120\sqrt{5}

, which simplifies to

24 \cdot 5^x = 120\sqrt{5}

. Dividing both sides by 24 isolates the exponential term:

5^x = 5\sqrt{5}

. Since

5\sqrt{5}

can be written as

5^1 \cdot 5^{0.5} = 5^{1.5}

, we have

5^x = 5^{1.5}

. Equating the exponents gives

x = 1.5

(or

3/2

Factor out

5^x

from the left side of the equation.

5^x(5^2 - 1) = 120\sqrt{5}

To apply exponent rules to rewrite

5^{x+2}

5^x \cdot 5^2

and then factor out the common term

5^x

Simplify the constant term inside the parentheses.

24 \cdot 5^x = 120\sqrt{5}

Evaluating

5^2 - 1 = 25 - 1 = 24

simplifies the coefficient of the exponential expression.

Divide both sides of the equation by 24.

5^x = 5\sqrt{5}

To isolate the exponential term

5^x

on one side of the equation.

Express the right side as a single power of 5.

5^x = 5^{1.5}

Using exponent rules where

5\sqrt{5} = 5^1 \cdot 5^{0.5} = 5^{1.5}

so that both sides have the same base.

Equate the exponents of the common base 5.

x = 1.5

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

Solving exponential equations by factoring and rewriting terms using a common base.

Estimated Time:1m 30s

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