3. Laws of Exponents

When multiplying two powers with the same base, add their exponents.

Example:

32 × 34 = 32+4 = 36 = 729

The Product of Powers Law states that when you have two exponents with the same base being multiplied, you simply add their exponents. For instance, if you have 2 raised to the power of 3 multiplied by 2 raised to the power of 4 (which is written as 2³ × 2⁴), you add the exponents (3 + 4) to get 2 raised to the power of 7 (2⁷). This results in 128.

Think of this like stacking boxes. If you have a stack of 3 boxes and you add another stack of 4 boxes, you end up with a total of 7 boxes. The exponent helps us quickly count the total number of boxes (or multiplications) without explicitly writing them all out.

When dividing powers with the same base, subtract the exponent of the denominator from that of the numerator.

Example:

56 ÷ 52 = 56−2 = 54 = 625

The Quotient of Powers Law tells us how to simplify expressions where the same base is divided. For instance, if you have 5 raised to the power of 6 divided by 5 raised to the power of 2 (5⁶ ÷ 5²), you subtract the exponents (6 − 2), yielding 5 raised to the power of 4 (5⁴), which equals 625.

Imagine you have 6 apples and you give away 2 apples. The total apples you have left can be thought of as subtracting the amount given away from your original amount, similar to how we subtract the exponents.

When raising a power to another power, multiply the exponents.

Example:

(23)2 = 23×2 = 26 = 64

According to the Power of a Power Law, when you take a power and raise it again to another power, you multiply the exponents. For example, in (2³)², you multiply the exponents 3 and 2 (3 × 2), resulting in 2 raised to the power of 6 (2⁶), which equals 64.

Imagine you are watching a movie trilogy. If each movie is 3 hours long and you plan to watch it twice, you would multiply the hours per movie by the number of times you plan to watch them. This is similar to how we multiply the exponents when applying the power of a power rule.

When raising a product to a power, apply the exponent to each factor.

Example:

(3×4)2 = 32 × 42 = 9×16 = 144

The Power of a Product Law indicates that when you have a product raised to a power, you apply that power to each factor within the parentheses. For instance, if you have (3×4)², you square both numbers: 3² and 4², giving you 9 and 16, respectively. Their product is therefore 144.

Think of it like baking cookies. If you have a recipe that requires 3 cups of sugar and you double the recipe, you would calculate 2 times each ingredient separately: 2×3 for sugar and then multiply the other ingredients—similar to applying the exponent to each part of the product.

When raising a quotient to a power, apply the exponent to both the numerator and denominator.

Example:

(2/5)3 = 2³/5³ = 8/125

The Power of a Quotient Law states that when you raise a fraction to a power, you apply the exponent to both the numerator and the denominator. For example, if you have (2/5)³, you cube both 2 and 5 separately, resulting in 2³ (which is 8) and 5³ (which is 125). Therefore, (2/5)³ equals 8/125.

Think of this as packing boxes for shipping. If each box can hold 2 items out of a total of 5 items (the full package), when you pack more boxes, you need to assess both the contents (the items, the numeral) and the overall box capacity (the denominator).