Lesson 47: Powers of 10

New Concept

When operating with exponents that share the same base, you can use these rules to simplify the expression.

Rules of Exponents

• To multiply: $a^x \cdot a^y = a^{x+y}$
• To divide: $\frac{a^x}{a^y} = a^{x-y}$
• To raise a power to a power: $(a^x)^y = a^{xy}$

What’s next

Next, you’ll apply these rules in worked examples. We will practice simplifying expressions and writing large numbers using powers of 10.

Property

When working with exponents: To multiply powers with the same base, add the exponents ($a^x \cdot a^y = a^{x+y}$). To divide, subtract the exponents ($\frac{a^x}{a^y} = a^{x-y}$). To raise a power to another power, you must multiply the exponents ($(a^x)^y = a^{xy}$).

Examples

$10^3 \cdot 10^4 = 10^{3+4} = 10^7$
$\frac{10^8}{10^2} = 10^{8-2} = 10^6$
$(10^3)^2 = 10^{3 \cdot 2} = 10^6$

Explanation

Think of exponents as a secret code for math! When you multiply numbers with the same base, just add the exponents. When you are dividing them, subtract the exponents. When raising a power to another power, you multiply the exponents together. It’s like a super simple shortcut for handling large numbers!

Property

To multiply a decimal number by a positive power of 10, we shift the decimal point to the right the number of places that is indicated by the exponent. This simple trick is an easy way to make the number get much bigger, much faster! It’s a very useful tool.

Examples

$46.235 \times 10^2 = 4623.5$
$25 \times 10^6 = 25,000,000$
$2.5 \text{ million} = 2.5 \times 10^6 = 2,500,000$

Explanation

Multiplying by a power of 10 is like giving your number a giant growth spurt! The exponent tells you exactly how many places you need to slide the decimal point over to the right. By doing this you are making your number instantly bigger, which is a neat trick to have.