Lesson 62: Multiplying Three or More Factors, Exponents

New Concept

An exponent is a number that shows how many times another number (the base) is to be used as a factor.

What’s next

Next, you’ll apply this concept to simplify expressions and use the area formula $A = s^2$ to solve problems.

To find the product of three or more numbers, multiply any two factors first, then multiply that result by the next factor. Continue this process until all factors are used. The order in which you multiply does not change the final product, thanks to the awesome Associative Property of Multiplication we learned about earlier.

$2 \times 5 \times 7 = (2 \times 5) \times 7 = 10 \times 7 = 70$
$4 \times 6 \times 5 = 4 \times (6 \times 5) = 4 \times 30 = 120$
$3 \times 2 \times 5 \times 10 = (3 \times 2) \times (5 \times 10) = 6 \times 50 = 300$

Think of it as a multiplication party! You can't dance with everyone at once, so you pick a partner, then the next, and so on. The best part? It doesn't matter who you multiply first. You can group numbers to make it easier, like finding a 10, to get the same answer.

An exponent is a number indicating how many times another number, the base, should be multiplied by itself. It appears as a smaller number to the upper right of the base. For example, in the expression five squared, the exponent is 2, which tells us to multiply 5 by itself, so $5 \times 5$.

$4^2 \text{ is read as 'four squared' and equals } 4 \times 4 = 16$.
$2^3 \text{ is read as 'two cubed' and equals } 2 \times 2 \times 2 = 8$.
$10^4 = 10 \times 10 \times 10 \times 10 = 10,000$.

An exponent is like a duplication command in a video game. The base is the character, and the exponent tells you how many copies to make through multiplication. A small exponent can lead to a surprisingly big result, making it a super-powerful tool for writing and calculating big numbers much more quickly!

An exponential expression is the complete unit, consisting of both a base and an exponent. It's a shorthand notation that instructs you to use the base as a factor the number of times specified by the exponent. This powerful tool helps us write very large numbers in a much more compact and manageable form.

The expression $5 \times 5 \times 5$ can be rewritten as $5^3$.
To simplify $5^2 + 10^2$, calculate each part first: $25 + 100 = 125$.
To simplify $3^3 - 2^2$, calculate each part first: $27 - 4 = 23$.

Think of an exponential expression as a recipe: the base is your ingredient, and the exponent tells you how many times to add it to the mix—through multiplication, of course! This whole package deal, like $7^4$, is a single instruction that tells a complete multiplication story without needing to write it all out.

The formula for the area of a square is $A = s^2$. This means the area ($A$) is found by taking the length of one side ($s$) and multiplying it by itself, or 'squaring' it. This formula is a direct and practical application of exponents, connecting the abstract concept to a real-world geometric shape.

To find the area of a square with 6-inch sides, use the formula: $A = s^2 = (6 \text{ in})^2 = 36 \text{ sq. in.}$
A square with sides of 10 meters has an area of $A = (10 \text{ m})^2 = 100 \text{ sq. m.}$
If a square patio is 8 feet long on each side, its area is $A = (8 \text{ ft})^2 = 64 \text{ sq. ft.}$

Ever wonder why an exponent of 2 is called 'squared'? This is it! To find the space inside a square, you just take its side length and use an exponent of 2. This single operation, $A = s^2$, quickly tells you how many little 1x1 squares can fit inside the bigger one.