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Powers of 10

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Basics on the topic Powers of 10

Powers of ten are when we multiply a number by ten a number of times. We can use indices as a short way of showing how many times we multiply by ten. This is a shorter way than writing the expanded version.

Transcript Powers of 10

The universe is an ever changing landscape as technology allows us to discover more galaxies. As of now, the observable universe measures about ninety-four billion light years with approximately trillions of galaxies inside of it. When studying the universe, we can help calculate these distances by using powers of ten. Our number system is based on groups of ten. Some maths problems use very large numbers, such as distances to other galaxies, which means we need a system to make them easier to work with. The powers of ten can be used to express or show large numbers. A power of ten is when we multiply ten by itself any number of times. First, we'll look at powers of ten by reviewing the expanded forms of multiplying by ten. They would be written out like this. What do you notice happens to the product each time we add another ten? By increasing the zeros in the product, it grows ten times larger each time. As you can see, writing our numbers in these expanded forms becomes more impractical as the number grows larger. As a result, the powers of ten use indices to express large numbers in a shorthand way. Indices are smaller numbers that we put in the upper right-hand corner of a base number. This is also known as the index or power. This number indicates how many times we must multiply the base number by itself. In this case, the base number is ten and the index is two. This number is read as ten to the power of two. In other words, it instructs us to multiply ten by itself two times to get one hundred. When we put all the forms of writing a number next to each other, we can see the connections. The index, two, is the same as the number of times the ten appears in this expression and it is the same as the number of zeroes in the number one-hundred. Let’s look at another example. Because the index is six, this is ten to the power of six. How many times will we multiply the ten? We'll write the ten six times and multiply them all together. This equals one million. Ten to the power of six means there are six zeroes in our answer. We can make a connection between powers of ten, expanded form expressions and the product. This chart shows us the patterns created by the powers of ten. As the indices increase, so do the number of tens multiplied and the number of zeros in the product. Let’s practise some powers of ten. What is ten to the power of seven written as an expanded expression? What is the product of this power? Ten million. Let's start with the expanded expression to determine the power of ten in this problem. How many times is the ten multiplied by itself here? Nine. How would we write this expression as a power of ten? Ten to the power of nine. What is ten to the power of nine? Ten to the power of nine is one billion. How many zeroes are in one billion? There are nine. To Summarise, powers of ten are a convenient way to express large numbers by combining the base number, ten, and the index or indices. The power of tens makes large numbers, such as distances to far away galaxies, easier to write and calculate.

Powers of 10 exercise

Would you like to apply the knowledge you’ve learnt? You can review and practice it with the tasks for the video Powers of 10 .
  • Define powers of ten.

    Hints

    The same number can be written in different ways but always has the same value.
    Example: $10^{2}$ is the same number as:

    • 10 x 10 (2 times)
    • 100 (2 zeros)
    • one hundred (written)

    When writing calculations with very large numbers, such as the distances between planets and galaxies in space, writing numbers, such as three hundred billion, with all the necessary zeros can be confusing, so we use exponents.

    There are 2 true statements.

    Solution

    TRUE

    • Powers of ten use exponents. Exponents are small numbers written in the upper right-hand corner of a base number.
    • $10^{3}$ means you must multiply 10 by 10 three times.
    FALSE
    • Powers of ten use despondents. Despodants are inactive numbers.
    • $10^{3}$ means you must multiply 10 by 3 three times.

  • Match powers of ten with their standard form number.

    Hints

    The exponent is the same as the number of zeros in standard form (or the number of zeros after 1).

    Use this chart to compare the same number written as a power of ten, an expression and in standard form.

    Solution
    • $10^{6}$ is the same number as 1,000,000.
    • $10^{10}$ is the same number as 10,000,000,000.
    • $10^{4}$ is the same number as 10,000.
    • $10^{3}$ is the same number as 1,000.
  • Connect powers of 10 with the expression or standard form.

    Hints

    A power of ten in standard form has as many zeros as the exponent.
    Example: $10^{10}$ = 10,000,000,000 (standard form, ten zeros)

    A power of ten as an expression multiplies 10 the same amount of times as the exponent.

    Example: 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 is the expression for $10^{10}$.

    Solution
    • $10^{3}$ = 10 x 10 x 10 (expression)
    • $10^{4}$ = 10,000 (standard form, 4 zeros)
    • $10^{6}$ = 10 x 10 x 10 x 10 x 10 x 10 (expression)
    • $10^{8}$ = 100,000,000 (standard form, 8 zeros)
  • Complete the table.

    Hints

    From left to right, the power of 10, expression, standard form and name are all the same number presented differently.

    The exponent tells you how many times to multiply 10 by 10, as well as how many zeros the standard form should have.

    The same number can be written in different ways but always has the same value. $10^{8}$ is the same number as 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 (8 times) and as 100,000,000 (8 zeros) and as one hundred million.

    Solution

    Above is the completed table.

  • Find the expression of the power of ten.

    Hints

    The same number can be written in different ways but always has the same value.
    Example: $10^{2}$ is the same number as:

    • 10 x 10 (2 times)
    • 100 (2 zeros)
    • one hundred (written)

    $10^{4}$ represents the expression of 10 multiplied by 10 4 times. (10 x 10 x 10 x 10).

    Solution

    $10^{7}$ is the same as 10 x 10 x 10 x 10 x 10 x 10 x 10 and is the same as 10,000,000.

  • Present the information in different forms.

    Hints

    The same number can be written in different ways but always has the same value. $10^{2}$ is the same number as 10 x 10 (expression) and as 100 (standard form) and as one hundred (name).

    Fill in the blanks carefully. Only write numbers (e.g. 100) when asked for numbers, and written names (e.g. one hundred) when asked for names.

    If you are unsure how something is spelled, these words may help.

    • One
    • Ten
    • Hundred
    • Thousand
    • Million
    • Billion
    • Trillion

    The exponent tells you how many times to multiply 10 by 10 for an expression, as well as how many zeros the standard form should have.

    If you have used commas in the number, check that these are in the correct position.

    Solution

    Above is the completed table for $10^{3}$ and $10^{6}$.