AMC 10 Step by Step
Log inSign up
Printable

2016 AMC 10A Problem 2

Problem onlyBack to the problem
2016 AMC 10A Problem 2AMC 10 Step by Step
Name:                             Date: September 13, 2026
Exponents, Logarithms & Radicals · difficulty 1 of 5 · about 1 min · position P1-10

For what value xx does 10x⋅1002x=1000510^{x}\cdot 100^{2x}=1000^{5} ?

  1. A)

    11

  2. B)

    22

  3. C)

    33

  4. D)

    44

  5. E)

    55

Answer

C

Key insight

Rewrite every base as a power of 10 so the equation becomes 10^{5x} = 10^{15}, then match exponents.

Solution

Express everything with base 1010: 100=102100 = 10^2 and 1000=1031000 = 10^3, so
1002x=(102)2x=104x,10005=(103)5=1015.100^{2x} = (10^2)^{2x} = 10^{4x}, \qquad 1000^5 = (10^3)^5 = 10^{15}.

The left side becomes 10x⋅104x=105x10^x \cdot 10^{4x} = 10^{5x}. Setting 105x=101510^{5x} = 10^{15} gives 5x=155x = 15, so x=3x = 3.

The answer is (C) 3\boxed{\textbf{(C)}\ 3}.

Why this works

When every number in an exponential equation is a power of one base, convert to that base and the equation collapses to a linear equation in the exponents. Keep the two rules straight: a power of a power multiplies exponents, a product of powers adds them.

Alternative approach

Test the choices: x=3x = 3 gives 103⋅1006=103⋅1012=1015=1000510^3 \cdot 100^6 = 10^3 \cdot 10^{12} = 10^{15} = 1000^5.

The trap

Writing 100^{2x} as 10^{2x+2} or 10^{2x} instead of 10^{4x}.

Common mistakes

  • Writing 100^{2x} as 10^{2x+2} or 10^{2x} instead of 10^{4x}.
  • Adding the exponents of the bases as if 1002x⋅10x100^{2x} \cdot 10^x were 1103x110^{3x} or similar nonsense with unlike bases.

Techniques

Set up the equation/formula and compute; no special trick needed