Multiplication Properties Of Exponents Homework Answers

When you get to the chapter on exponents, your teacher is going to be assigning lots and lots of homework of problems. Mostly because they'll probably be short. Well you got to be really careful though because a lot of times when I assign lots of exponents problems, students do most of them incorrectly because they try to rush through. Please be careful when you're doing these exponent problems there's lots of places to make mistakes. One place students make mistakes is they to memorize all of these exponents properties that we'll go over in a second and then they end up getting them confused in their brain. So I am going to go over these but I'm going to ask that you don't memorize them unless you think you're like an A level student. It's okay if you're a good memorizer and you're like a B or C student, you can also try to memorize these. But be really careful a lot of errors happen when students are just going through carelessly and they make silly mistakes because they thought they memorized the properties.
And one more thing before we go over these, these are just the properties of multiplying and dividing with exponents, there are other properties have to do with exponents that are zero and negative exponents, we'll get to those in another video. But for now let's just check this out I'm going to do these both using variables and also using numbers you guys can get a sense of what I mean. x to the n, to the m power is equal to the x to the n multiplied by m. Here's what it means if you're using numbers, 3 squared to the fourth power would be equal to 3 to the eighth power. That's what it would look like in terms of numbers, and we'll get into why that is when we start doing more and more practice problems.
Here's another property, if I have 3 times 2 to the fourth power like xy to the m, it's equal to x to the m times y to the m. So that would be 3 to the fourth times 2 to the fourth, it's almost like that little exponent distributes on the each of these pieces in the base as long as they are in parenthesis. Here is another property are you getting tired of trying to memorize them yet, don't worry with memorizing. You can work on that and I'll show you that when we do problems. x to the n times x to the m is equal to x to the n plus m. What does that mean? It means this, if you have the same base, like 3 to the 1 times 3 to the second power see how all those are the same base but different exponents that can be simplified by adding those guys 1 plus 2.
Again if you're not a good memorizer don't be memorizing these, we're going to go over them and make them make more sense in just a second. Next one x to the n divided by x to the m is equal to the difference of the exponents. So if I had 3 to the fourth power on top of the 3 squared that would be equal to the same thing as 3 to the 4 take away 2 or 3 squared.
Last but not least if you have a fraction raised to an exponent it's kind of like this exponent gets distributed on to both pieces of that base. Like 3 halves to the fourth is the same thing as 3 to the fourth over 2 to the fourth. So I'm just going to say this one last time I know I've already said it like lots of times, if you're not someone whose a good memorizer don't even try to memorize these because you're going to get them jumbled in your head. It's always better to write out every number and get it correct than to do it shortcut method easy in memorizing but you get it wrong.

In earlier chapters we introduced powers.

$$x^{3}=x\cdot x\cdot x$$

There are a couple of operations you can do on powers and we will introduce them now.

We can multiply powers with the same base

$$x^{4}\cdot x^{2}=\left (x\cdot x\cdot x\cdot x \right )\cdot \left ( x\cdot x \right )=x^{6}$$

This is an example of the product of powers property tells us that when you multiply powers with the same base you just have to add the exponents.

$$x^{a}\cdot x^{b}=x^{a+b}$$

We can raise a power to a power

$$\left ( x^{2} \right )^{4}= \left (x\cdot x \right )\cdot \left (x\cdot x \right ) \cdot \left ( x\cdot x \right )\cdot \left ( x\cdot x \right )=x^{8}$$

This is called the power of a power property and says that to find a power of a power you just have to multiply the exponents.

When you raise a product to a power you raise each factor with a power

$$\left (xy \right )^{2}= \left ( xy \right )\cdot \left ( xy \right )= \left ( x\cdot x \right )\cdot \left ( y\cdot y \right )=x^{2}y^{2}$$

This is called the power of a product property

$$\left (xy \right )^{a}= x^{a}y^{a}$$

As well as we could multiply powers we can divide powers.

$$\frac{x^{4}}{x^{2}}=\frac{x\cdot x\cdot {\color{red} \not}{x}\cdot {\color{red} \not}{x}}{{\color{red} \not}{x}\cdot {\color{red} \not}{x}}=x^{2}$$

This is an example of the quotient of powers property and tells us that when you divide powers with the same base you just have to subtract the exponents.

$$\frac{x^{a}}{x^{b}}=x^{a-b},\: \: x\neq 0$$

When you raise a quotient to a power you raise both the numerator and the denominator to the power.

$$\left (\frac{x}{y} \right )^{2}=\frac{x}{y}\cdot \frac{x}{y}=\frac{x\cdot x}{y\cdot y}=\frac{x^{2}}{y^{2}}$$

This is called the power of a quotient power

$$\left (\frac{x}{y} \right )^{a}=\frac{x^{a}}{y^{a}},\: \: y\neq 0$$

When you raise a number to a zero power you'll always get 1.

$$1=\frac{x^{a}}{x^{a}}=x^{a-a}=x^{0}$$

$$x^{0}=1,\: \: x\neq 0$$

Negative exponents are the reciprocals of the positive exponents.

$$x^{-a}=\frac{1}{x^{a}},\: \: x\neq 0$$

$$x^{a}=\frac{1}{x^{-a}},\: \: x\neq 0$$

The same properties of exponents apply for both positive and negative exponents.

In earlier chapters we talked about the square root as well. The square root of a number x is the same as x raised to the 0.5th power

$$\sqrt{x}=\sqrt[2]{x}=x^{\frac{1}{2}}$$


Video lesson

Simplify the following expression using the properties of exponents

$$\frac{( 7^{5}) ^{10}\cdot 7^{200}}{\left ( 7^{-2} \right )^{30}}$$

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