4.5 × 10⁻⁴. Moving decimal point and negative exponents.

Express 0.00045 in Standard Form

Question

Express 0.00045 in standard form.

Solution — Step by Step

Standard form (also called scientific notation) means writing a number as $a \times 10^n$ , where $1 \leq a < 10$ and $n$ is an integer. We need to shift the decimal point so exactly one non-zero digit sits to its left.

In 0.00045, the first non-zero digit is 4. We want the decimal point to land between 4 and 5, giving us 4.5.

Starting from 0.00045, we move the decimal point 4 places to the right to get 4.5.

0.00045  →  0.0045  →  0.045  →  0.45  →  4.5
  (1)         (2)        (3)       (4)

Moving the decimal to the right means the original number is smaller than 4.5. So we compensate with a negative exponent. 4 places to the right → power of $-4$ .

0.00045 = 4.5 \times 10^{-4}

Why This Works

Every time we multiply by 10, the decimal shifts one place to the right. So multiplying by $10^4$ shifts it 4 places right. To keep the number equal, we must also divide by $10^4$ — which is the same as multiplying by $10^{-4}$ .

That’s the core logic: $4.5 \times 10^{-4} = \dfrac{4.5}{10000} = 0.00045$ . The negative exponent is just a compact way of saying “divide by this power of 10.”

Quick check: Negative exponent → small number (less than 1). Positive exponent → large number (greater than 10). If your answer doesn’t match this pattern, something went wrong.

Alternative Method

Work backwards from the definition.

We know $10^{-4} = \dfrac{1}{10000} = 0.0001$ . So:

4.5 \times 10^{-4} = 4.5 \times 0.0001 = 0.00045 \checkmark

Some students find it easier to first guess the coefficient (4.5 here) and then verify by computing the product. This is especially useful when checking your work in a time-pressured board exam.

Common Mistake

Wrong exponent sign — writing $4.5 \times 10^{4}$ instead of $4.5 \times 10^{-4}$ .

Students count the 4 jumps correctly but forget that moving the decimal right (to make the number bigger) needs a negative power to bring it back down. A quick sanity check: $10^4 = 10000$ , so $4.5 \times 10^4 = 45000$ . That’s nowhere near 0.00045 — the wrong sign inflates the number by a factor of $10^8$ .

Final Answer: $0.00045 = \mathbf{4.5 \times 10^{-4}}$

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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