Powers of 10: How Moving the Decimal Changes Everything

A power of 10 is the result of multiplying 10 by itself a set number of times. Powers of 10 show up in measurement, science, money, and the place value system students use every day.

They also answer a question that comes up in Grades 4 through 7 when students multiply or divide by 10, 100, or 1,000 and wonder why the decimal moves.

Today, our Mathnasium tutors walk you through the most common powers of 10, how they connect to the place value chart, and how multiplying or dividing by a power of 10 moves the decimal.

How Powers of 10 Work 

As we mentioned above, a power of 10 is the result of multiplying 10 by itself a set number of times. The small number written above and to the right of the 10 is called the exponent. It tells you how many times 10 is multiplied by itself.

For example, 10³ means 10 × 10 × 10, which equals 1,000.

The table below shows the six most common powers of 10, how they are calculated, and what they look like in the real world.

Two things stand out in this table. First, each time the exponent increases by 1, a zero is added to the result. 10³ has three zeros. 10⁶ has six zeros.

Second, the results grow by a factor of 10 each time. Going from 10¹ to 10² does not add 10 to the result. It multiplies it by 10, turning 10 into 100. Going from 10² to 10³ multiplies 100 by 10 to get 1,000. 

Each row is ten times the size of the row before it, not ten more than it.

The Decimal Point and Place Value

So far, we've looked at powers of 10 that grow larger, like 100 and 1,000. But powers of 10 also work in reverse to represent parts of a whole, which brings us straight to our place value system and decimals.

The chart below shows how every column in the place value system connects to a power of 10. The decimal point is fixed at the ones column and never moves.

Follow the chart from ones upward. 

• Ones become tens. 

• Tens become hundreds. 

• Hundreds become thousands. 

Each step up makes the value 10 times bigger.

That means moving one column up is the same as multiplying by 10. Moving two columns up is the same as multiplying by 100, which is 10². Moving three columns up is the same as multiplying by 1,000, which is 10³.

Now follow the chart from ones downward. 

• Ones become tenths. 

• Tenths become hundredths. 

Each step down makes the value 10 times smaller.

We know that the decimal point stays fixed at the ones column. What changes is which column each digit occupies. 

When digits move far enough downward past the ones column, they become decimal fractions. That movement is what makes the decimal point appear to shift when multiplying or dividing by a power of 10.

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Multiplying by a Power of 10: Why the Decimal Appears to Move Right

Multiplying a number by a power of 10 shifts every digit the same number of columns toward larger values, equal to the exponent. 

Because the decimal point stays fixed at the ones column, those shifting digits make the decimal point appear to move right.

Look at what happened to the digits in 0.25. The digit 2 sits in the tenths column. The digit 5 sits in the hundredths column.

• 0.25 × 10¹ = 2.5

• 0.25 × 10² = 25

• 0.25 × 10³ = 250

Now look at where the decimal point is in each answer. In 0.25, it sits before the 2. In 2.5, it sits after the 2. In 25, the digits have moved far enough left that the decimal point is no longer visible, but it is still there after the 5. The digits moved left. The decimal point stayed fixed at the ones column throughout.

The shortcut students often use is to just add zeros. Try it with 2.5 × 10². Adding zeros gives 2.500, which is the same as 2.5. That is not 250. The shortcut only works for whole numbers, not for decimals.

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Dividing by a Power of 10: Why the Decimal Appears to Move Left

Dividing by a power of 10 works the same way as multiplying, but in the other direction. Instead of shifting toward larger values, every digit shifts the same number of columns toward smaller values.

Let's start with a whole number to see the shift clearly. 

Look at the digit 2 in 2,500. It starts in the thousands column. After dividing by 10³, it has moved three columns toward smaller values and landed in the ones column. The digit 5 started in the hundreds column and moved to the tenths column.

• 2,500 ÷ 10¹ = 250

• 2,500 ÷ 10² = 25

• 2,500 ÷ 10³ = 2.5

Did you notice what happened when the 5 moved past the ones column? It became a decimal fraction. That is why the answer is 2.5 and not just 2. The digit did not disappear. It moved past the decimal point into the tenths column.

Now let's try a number that already has a decimal, so the leftward movement is easier to see.

• 25.8 ÷ 10¹ = 2.58

• 25.8 ÷ 10² = 0.258

• 25.8 ÷ 10³ = 0.0258

Watch what happens to the decimal point in each answer. In 25.8, the decimal sits between the tens and the ones. After dividing by 10¹, it moves one column to the left, sitting between the ones and the tenths. After dividing by 10², it moves two columns to the left. After dividing by 10³, it moves three columns to the left.

The decimal point itself never moved. The digits shifted toward smaller values, and the decimal point marks where the ones column is. As the digits moved further left, the decimal point appeared to follow.

This is what happens when converting cents to dollars or inches to feet. The digits shift toward smaller values, and the decimal point appears to move left.

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Mathnasium tutors help students connect powers of 10 to the place value system they already know.

How Mathnasium Helps Students Master Powers of 10

Mathnasium is a math-only learning center that helps K–12 students of all skill levels truly understand math, not just follow steps. 

We help students build a deep understanding of a math concept, including powers of 10.

To help students reach this level of understanding, we use the Mathnasium Method™, our proprietary teaching approach.

Here’s how it works.

Every student begins with a diagnostic assessment that tells us exactly where their understanding is solid and where the confusion began. For a student struggling with powers of 10, that might mean going back to place value foundations, or it might mean reintroducing the concept of digit movement in a way that finally makes the decimal point make sense. From there, we build a personalized learning plan tailored to their specific needs.

Our specially trained tutors use a combination of verbal, visual, mental, tactile, and written techniques to make concepts like powers of 10 feel logical rather than arbitrary. We teach for understanding, which means students leave each session knowing not just what to do but why it works.

Sessions take place in a caring and fun group environment where students feel comfortable asking questions and working through confusion at their own pace.

And the results speak for themselves:

• 94% of parents report an improvement in their child's math skills and understanding

• 93% of parents report their child's improved attitude toward math after attending Mathnasium

• 90% of students saw an improvement in their school grades

With over 1,100 learning centers across North America, there is likely a Mathnasium close to you.

For families in and around Cave Creek, Mathnasium of Cave Creek brings that same approach close to home, with specially trained tutors who help students connect the procedures they are learning in class with lasting mathematical understanding.

If your child can follow the steps but finds it hard to explain why the decimal point moves or apply the concept in an unfamiliar problem, a free diagnostic assessment is a great place to start. From there, we create a personalized learning plan that helps your child develop the understanding and fluency they need next.

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