Product of Powers
Here's the formula: (x^a)(x^b) = x^(a + b). When you multiply exponentials with the same base (notice that x and x are the same base), add their exponents (or powers).
Let me show you how that works. Let's say I have (x^2)(x^3). Well, x^2 is x times x, and x^3 is x times x times x. When we add all those xs up, we get x^5, which is the same thing as adding 3 + 2.
Power to a Power
The Power of a Product property , We can see from the formula we have (x^a)^b. When you have a power to a power, you multiply the exponents (or powers). Let me show you how this one works.
If I have (x^2)^4, which would be x^2 multiplied four times, or x^2 times x^2 times x^2 times x^2. Once again, we add all the exponents and get x^8, and x^8 is the same as x^(2 * 4), which is 8. Not too bad, right?
Quotient of Powers
Remember, 'quotient' means 'division'.' The formula says (x^a) / (x^b) = x^(a - b). Basically, when you divide exponentials with the same base, you subtract the exponent (or powers).
Let me show you how this one works. Let's say I had (x^4) / (x^3). In the top (or numerator), we have x times x times x times x. In the bottom (or denominator), we have x times x times x. Hopefully, you remember that x divided by x is 1, so the xs cancel. So, x divided by x is 1, x divided by x is 1, and x divided by x is 1. So, when we cancel them, what are we left with? That's right: x^1, or just x. So (x^4) / (x^3) is just x^(4 - 3), which is x^1.
The Power of a Quotient Property
Power of a Product
The formula says (xy)^a = (x^a) and (y^a). When you have a product of a power, you give each base its own exponent. Think about it as distribution putting the exponent with each base.
Let me show you how this one works. Let's say we had (xy)^2. That means we take xy and multiply it twice that means xy times xy. Well, that would give us two xs, or x^2, and two ys, or y^2. That is the same as if I distributed 2 to the x, getting x^2, and 2 to the y, getting y^2.
Power of a Quotient
When we look at this formula, we have x/y, or a fraction raised to the a power, this gives us (x^a) / (y^a). When you have a quotient to a power, you give each base its own exponent. We think of it as the exponent being distributed to each part of the fraction, just like the last one, power of a product.
Let me show you how this one works. Let's say I have (x / y)^3. Remember, that means I'm going to take x / yand multiply it three times. That would be x / y times x / y times x / y. If we look at the top (or the numerator), we have x times x times x, or x^3. If we look at the bottom (or denominator), we have y times y times y, or y^3. That would give us (x^3) / (y^3), which is basically distributing 3 to the x and y .
Lesson Summary
Let's take a final minute to review the five properties.
- Product of a Power: When you multiply exponentials with the same base, you add their exponents (or powers).
- power to a Power: When you have a power to a power, you multiply the exponents (or powers).
- Quotient of Powers: When you divide exponentials with the same base, you subtract the exponents (or powers).
- Power of a Product: When you have a product of a power, you give each base its own exponent (or distribute the exponent to each base).
- Power of a Quotient: When you have a quotient to a power, you give each base its own exponent.
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