# What is 10 12 4

### Powers of ten with positive exponents

Powers of ten with a positive exponent are used,

to write large numbers like $$ 1 000 $$ or $$ 10 000 $$ more clearly.

The basis is always $$ 10 $$. That is always the same as the number of zeros.

$$1$$ $$=10$$

$$1$$ $$=10$$

$$ 1 $$ $$ = 10 $$ $$ 1 $$ thousand

$$1$$ $$=10$$

$$1$$ $$=10$$

$$ 1 $$ $$ = 10 $$ $$ 1 $$ million

…

$$ 1 $$ $$ = 10 $$ $$ 1 $$ billion

…

Have you noticed? Some units have prefixes that refer to the powers of ten, e.g. **Mega**byte.

description | Power of ten | example |
---|---|---|

Hecto ... | $$10^2$$ | Hectoliters |

Kilo… | $$10^3$$ | kilometre |

Mega ... | $$10^6$$ | megabyte |

Giga ... | $$10^9$$ | Gigaherz |

Powers of ten are powers with:

- the base $$ 10 $$
- integer exponents

Examples: $$ 10 ^ 2 $$, $$ 10 ^ -3 $$

$$ 10 ^ n $$ means a $$ 1 $$ with $$ n $$ zeros

### Examples

### 1) Convert to powers of ten

**Task:** Represent the number $$ 10 $$ $$ 000 $$ $$ 000 $$ $$ 000 $$ with a power of ten.

**Step 1:** Count the zeros of the number.

They are zeros.

**2nd step:** Use $$ 10 $$ as the base and the number of zeros as.

$$10$$ $$000$$ $$000$$ $$000=10$$

### 2) Calculate powers of ten

**Task:** Write the number $$ 10 ^ 12 $$ without a power of ten.

**Step 1:** Make a note of that.

The exponent is.

**2nd step:** Add zeros to the $$ 1 $$ according to the exponent.

$$ 1 $$ with zeros: $$ 1 $$

**More positive** Exponent = zeros **right** Add

### Separate powers of ten with positive exponents

With separated powers of ten, you can make numbers like

Write $$ 200 $$ $$ 000 $$ or $$ 34 $$ $$ 000 $$ $$ 000 $$ more clearly.

separated power of ten

$$ uarr $$

$$3,4 * 10^7$$

$$ darr $$

Number between $$ 1 $$ and $$ 10 $$

### Examples:

$$200$$ $$000$$ $$= 2 * 10^5$$

$$34$$ $$000$$ $$000$$ $$= 3,4 * 10^7$$

The number before the power of ten is between $$ 1 $$ and $$ 10 $$, but the $$ 10 $$ is no longer allowed.

$$ a * 10 ^ z $$ $$ z $$ from $$ ZZ $$ and $$ 1≤ a <10 $$

*kapiert.de*can do more:

- interactive exercises

and tests - individual classwork trainer
- Learning manager

### Examples

### 1) Convert to separate powers of ten

**Task:** Write the number $$ 56030000 $$ with the power of ten separated.

**Step 1:** Put a after the first digit of the number.

$$5$$$$6030000$$

**2nd step:** Count the ones after the decimal point.

The number of digits is the

the power of ten. They are digits.

So $$ 10 $$.

**3rd step:** Delete all trailing zeros and

multiply by the power of ten.

$$5,603 * 10$$

### 2) Calculate separate powers of ten

**Task:** Write the number $$ 2.163 * 10 ^ 4 $$ without the power of ten.

**Step 1:** Make a note of the power of ten.

The exponent is.

**2nd step:** Move the comma by the value

of the exponent to the right.

Move the comma one places to the right.

$$21630$$

One also says: represent the number in scientific notation.

$$ 10 ^ n $$ shifts the comma by n places to the right

$$ 2.163 = 2.16300000… $$ You can add trailing zeros.

### Powers with the formula editor

This is how you enter powers in kapiert.de with the formula editor (from approx. 1:30 in the video):

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