"What Is the Value of (0.3)⁵/(0.3)³?" — GMAT® Worked Solution

Source: Official Guide for GMAT® Review, 11th Edition

$\frac{(0.3)^5}{(0.3)^3}$

(A) $0.001$

(B) $0.01$

(C) $0.09$

(D) $0.9$

(E) $1$

Try it before reading on.

Step 1: Divide Powers With the Same Base by Subtracting Exponents

Both the numerator and the denominator are powers of the same base, 0.3, so the exponent rule for division applies: subtract the denominator's exponent from the numerator's.

$\frac{(0.3)^5}{(0.3)^3} = (0.3)^{5-3} = (0.3)^2$

It's worth knowing why the subtraction is legal. Five factors of 0.3 sit on top:

$\frac{0.3 \times 0.3 \times 0.3 \times 0.3 \times 0.3}{0.3 \times 0.3 \times 0.3}$

Three factors on top cancel against the three on the bottom, and two factors survive. That's all the rule does: it counts the survivors.

Step 2: Compute (0.3)² With Decimal Long Multiplication

The remaining work is multiplication, and this is where the most popular wrong answer lives. The long multiplication setup:

    0.3
  × 0.3
  ------
      9     3 × 3, decimals set aside
    0.09    decimal shifted two places left

Two factors, one decimal place each, means two decimal places in the product. Start from the 9, count the decimal places that got set aside, and shift the decimal two places to the left:

$0.3 \times 0.3 = 0.09$

The answer is (C).

Where the Wrong Answers Come From

Choice (D) is 0.9, and it's the most common incorrect answer on this question. That's the correct product with the decimal moved one place instead of two. The multiplication itself is right, which is why the slip survives a mental double-check that only re-runs the multiplication: the fix is counting the decimal places in writing, one from each factor.

Choice (E) is what happens when the bases get canceled the way fraction terms cancel, as if $\frac{(0.3)^5}{(0.3)^3}$ simplified to something over itself and landed on 1. Powers of the same base don't work that way; subtracting the exponents in Step 1 is the legal version of that cancellation. The remaining choices catch other slips in the same two skills, exponents and decimal placement.

Why This Problem Matters

The exponent shortcut is memorization, the decimal multiplication is execution, and this question needs both for about ten seconds of work. Each half fails differently: the shortcut has to be known cold, and the decimal count has to be written down to be trusted.

If decimal placement was the step that got you here, that's a common story, and it's worth telling straight: decimal long multiplication is a skill plenty of working adults haven't touched since school, and some of us learned it for the first time during GMAT® prep. It tends to come back quickly with a few drills, and the decimal long multiplication episode of The GMAT® Strategy Podcast walks through the full procedure.

For anyone shaky on the exponent shortcut, the intermediate step is writing all five factors out longhand and crossing off the three that cancel, physically, on the page. It's slower, and it's a temporary device: after a couple of weeks of it, the subtraction rule tends to stick on its own. Scratch work doesn't get graded; only the score does.

Ready for the next one? "In a Horticultural Experiment, 200 Seeds Were Planted in Plot I..." — GMAT® Worked Solution.

Want the full strategy? Read: GMAT® Setup Habits: The Question-First System

From Episode 13 of Real GMAT® Problems (The GMAT® Strategy Podcast).