If x is a positive integer such that \sqrt{4^{x+1} \cdot 5^{2x} + 4^x \cdot 5^{2x+1}} = 30… — GRE General Test

x

is a positive integer such that

\sqrt{4^{x+1} \cdot 5^{2x} + 4^x \cdot 5^{2x+1}} = 30,000

what is the value of

x

Answer: 4

The value of

x

is 4.

Factoring the common exponential term

4^x \cdot 5^{2x}

inside the radical yields

\sqrt{(4^x \cdot 5^{2x})(4 + 5)} = \sqrt{9 \cdot 4^x \cdot (5^2)^x} = \sqrt{9 \cdot (4 \cdot 25)^x} = \sqrt{9 \cdot 100^x} = \sqrt{9 \cdot 10^{2x}} = 3 \cdot 10^x

. Setting

3 \cdot 10^x = 30,000

gives

10^x = 10,000 = 10^4

, which means

x = 4

Separate the addition in exponents using exponent rules.

4^{x+1} \cdot 5^{2x} = 4^x \cdot 4^1 \cdot 5^{2x}

and

4^x \cdot 5^{2x+1} = 4^x \cdot 5^{2x} \cdot 5^1

Applying the product rule of exponents

a^{m+n} = a^m \cdot a^n

prepares terms for factoring.

Factor out the common expression

4^x \cdot 5^{2x}

from the sum inside the radical.

4^{x+1} \cdot 5^{2x} + 4^x \cdot 5^{2x+1} = (4^x \cdot 5^{2x})(4 + 5) = 9 \cdot 4^x \cdot 5^{2x}

Factoring converts the sum under the square root into a single product.

Combine terms with powers into base 10.

4^x \cdot 5^{2x} = 4^x \cdot (5^2)^x = 4^x \cdot 25^x = (4 \cdot 25)^x = 100^x = 10^{2x}

Using power of a power

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

and power of a product

a^n b^n = (ab)^n

simplifies the expression into powers of 10.

Take the square root of the simplified product.

\sqrt{9 \cdot 10^{2x}} = \sqrt{9} \cdot \sqrt{10^{2x}} = 3 \cdot 10^x

Applying the product rule for radicals

\sqrt{ab} = \sqrt{a}\sqrt{b}

and halving the exponent

(10^{2x})^{1/2} = 10^x

Equate the simplified expression to 30,000 and solve for

x

3 \cdot 10^x = 30,000 \implies 10^x = 10,000 \implies 10^x = 10^4 \implies x = 4

Dividing both sides by 3 isolates

10^x

, and matching exponential bases gives

x = 4

Exponent Rules and Radical Simplification

Estimated Time:2m 0s

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