If 5^{x} - 5^{x-2} = 600\sqrt{5}, what is the value of x? — GMAT

If $5^{x} - 5^{x-2} = 600\sqrt{5}$ , what is the value of $x$ ?

Answer: 4.5

The value of

x

is 4.5.

Factoring out

5^{x-2}

converts the left side into

5^{x-2}(25 - 1) = 24 \cdot 5^{x-2}

. Dividing

600\sqrt{5}

by 24 gives

25\sqrt{5}

, which equals

5^{2.5}

. Setting

x - 2 = 2.5

gives

x = 4.5

Factor out the smallest power of 5, which is

5^{x-2}

, from the left side of the equation.

5^{x-2}(5^2 - 1) = 600\sqrt{5}

Factoring isolates the constant multiplier from the variable power term.

Calculate the numerical value inside the parentheses.

24 \cdot 5^{x-2} = 600\sqrt{5}

5^2 - 1 = 25 - 1 = 24

Divide both sides of the equation by 24.

5^{x-2} = 25\sqrt{5}

Isolating

5^{x-2}

gives

\frac{600\sqrt{5}}{24} = 25\sqrt{5}

Rewrite

25\sqrt{5}

as a single exponential expression with base 5.

5^{x-2} = 5^2 \cdot 5^{0.5} = 5^{2.5}

Using the product rule for exponents,

5^2 \cdot 5^{1/2} = 5^{2 + 0.5} = 5^{2.5}

Equate the exponents since the bases are identical.

x - 2 = 2.5 \implies x = 4.5

For any positive base

b \neq 1

, if

b^m = b^n

, then

m = n

Exponents, Roots, and Powers of Integers

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