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### Course: 8th grade (Illustrative Mathematics)>Unit 7

Lesson 3: Lesson 8: Combining bases

# Exponent properties with parentheses

Learn two exponent properties: (ab)^c = (a^c)*(b^c) and (a^b)^c = a ^ (b*c). See WHY they work and HOW to use them. In other words, multiplying two numbers, then raising the product to an exponent is the same as raising each number to that exponent and then multiplying. Raising a number to an exponent and then to another exponent equals raising the base to the product of the two exponents. Created by Sal Khan.

## Want to join the conversation?

• a^3 * a^3 = a^6 What happens when you have a^3 + a^3?
• if you take a^3 and say it is equal to an arbitrary letter b (a^3 = b)
you can simply say that b + b = 2b which leads to 2a^3 since b is equal to a^3
• can a exponent be a negative fraction or mixed number?
• yes, exponents can be negative fraction or mixed number
• Can an exponent be a fraction or a decimal? Or must it be a whole number?
• Yes, exponents can be in fraction or a decimal. Although decimal exponents are less common and you will see exponents as fractions almost all the time.
• Can negative numbers have a negative exponent?
• Yes. For example, (-3)^(-4) = 1/[(-3)^4] = 1/81, and (-4)^(-3) = 1/[(-4)^3] = 1/(-64) = -1/64.
Have a blessed, wonderful day!
• can you use exponents to show exponents? For example 3 to the 6th power and the 6th power is shown as 6 to the 1st power?
• yes, you can, 3^6^1 = 3^6 = 729, for a better example, 2^2^3 = 2^8 = 256, but (2^2)^3 = 4^3 = 64
• Isn't it easier just to multiply the exponent on the outside by the one in the parenthesis?

example:
(2^3)^2

2*3=6

so 2^6
• you can do it the other way as well if need be but I agree
• can a exponent be a fraction
• Great question! Yes, exponents can be fractions! When we take a number to a fractional power, we interpret the numerator as a power and the denominator as a root. For example:

25^(1/2) = [sqrt(25)]^1 = sqrt(25) = 5.
16^(3/4) = [4throot(16)]^3 = 2^3 = 8.
1,000^(4/3) = [cuberoot(1,000)]^4 = 10^4 = 10,000.