When $n$ is positive, we multiply $10$ by itself $n$ times,
\n\\[\\text{e.g. } 10^3 = 10 \\times 10 \\times 10 = 1000 \\text{ .}\\]
\nWhen $n$ is negative, we can think of $10^{-n}$ as $\\frac{1}{10^{n}}$,
\n\\[\\text{e.g. } 10^{-3} = \\frac{1}{10^3} = \\frac{1}{1000} = 0.001\\text{ .}\\]
\nWhen $n = 0$:
\n\\[10^{0} = 1 \\text{ .}\\]
\nGenerally, we can think of $10^n$ as a number in standard form $1 \\times 10^n$. Then $n$ always tells us the number of decimal places to move the decimal point in $1.0$, for example
\n\\[10^{-3} = 1.0 \\times 10^{-3} \\text{ and since } n = - 3 \\text{, we go } 3 \\text{ places back as follows: } 1.0 ⇒ 0.1 ⇒ 0.01 ⇒ 0.001 \\text{ .}\\]
\nA complete table of powers of ten for $n$ from $-6$ to $6$ is:
\n| $n$ | \n$10^n$ | \n
|---|---|
| $-6$ | \n$0.000001$ | \n
| $-5$ | \n$0.00001$ | \n
| $-4$ | \n$0.0001$ | \n
| $-3$ | \n$0.001$ | \n
| $-2$ | \n$0.01$ | \n
| $-1$ | \n$0.1$ | \n
| $0$ | \n$1$ | \n
| $1$ | \n$10$ | \n
| $2$ | \n$100$ | \n
| $3$ | \n$1000$ | \n
| $4$ | \n$10000$ | \n
| $5$ | \n$100000$ | \n
| $6$ | \n$1000000$ | \n
Powers of ten can be useful while working with standard index numbers. Fill in the following table of powers of ten:
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", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "{-n-1}", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "{-n-1}", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "10^{-n+1}", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "10^{-n+1}", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "10^{n}", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "10^{n}", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "{n+2}", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "{n+2}", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "prompt": "\n| $n$ | \n$10^n$ | \n
|---|---|
| [[0]] | \n$\\var{10^(-n-1)}$ | \n
| $\\var{-n+1}$ | \n[[1]] | \n
| $0$ | \n$1$ | \n
| $1$ | \n$10$ | \n
| $\\var{n}$ | \n[[2]] | \n
| [[3]] | \n$\\var{10^(n+2)}$ | \n