Question

Tutorial: Exponents

Answer 1

\(\bf\huge Basics:\)

\(\large\color{red}{Base \rightarrow x}^{\color{blue}{a \leftarrow Exponent}}\)

Exponents represent how many times the base is multiplied by itself.

Example:

\(2^5\) represents 2 being multiplied by itself 5 times. We can write this like \(2 \cdot 2 \cdot 2 \cdot 2\cdot 2 = 32\).

Remember:

All numbers have exponents, for example:

\(5\) has an exponent of \(1\), even though we don’t write it.

\(\bf\huge Rules:\)

We cannot simplify exponents without like bases, unless we simplify them as whole numbers.

\(\bf\small Multiplication~ Rule:\)

When multiplying exponents with like bases, keep the base and add the exponents.

\(x^\color{red}{a} \cdot x^\color{blue}{b} \rightarrow x^{\color{red}{a}+\color{blue}{b}}\)

Examples:

\(5^\color{red}{2} \cdot 5^\color{blue}{6} \rightarrow 5^{\color{red}{2}+\color{blue}{6}} \rightarrow 5^8\)

\(4 \cdot 4^\color{blue}{3} \rightarrow 4^{\color{red}{1}+\color{blue}{3}} \rightarrow 4^4\)

\(\bf\small Division ~Rule:\)

When dividing exponents with like bases, keep the base and subtract the exponents.

\(\dfrac{x^\color{red}{a}}{x^\color{blue}{b}} \rightarrow x^{\color{red}{a}-\color{blue}{b}}\)

Examples:

\(\dfrac{3^\color{red}{2}}{3^\color{blue}{3}} \rightarrow 3^{\color{red}{2}-\color{blue}{3}} \rightarrow 3^{-1}\)

\(\dfrac{8^\color{red}{6}}{8^\color{blue}{2}} \rightarrow 8^{\color{red}{6}-\color{blue}{2}} \rightarrow 8^{4}\)

\(\dfrac{2^\color{red}{2}}{2} \rightarrow 2^{\color{red}{2}-\color{blue}{1}} \rightarrow 2^{1}\)

\(\bf\small Outside~ Exponent:\)

When you have an exponent outside of a parenthesis, you multiply it with the exponents inside the parenthesis.

\((x^\color{red}{a})^\color{blue}{b} \rightarrow x^{\color{red}{a} \cdot \color{blue}{b}}\)

Examples:

\((3^\color{red}{2})^\color{blue}{4} \rightarrow 3^{\color{red}{2} \cdot \color{blue}{4}} \rightarrow 3^8\)

\((x^\color{red}{2}y^\color{blue}{3})^\color{lime}{2} \rightarrow x^{\color{red}{2 \cdot \color{lime}{2}}} y^{\color{blue}{3} \cdot \color{lime}{2}} \rightarrow x^4y^6\)

\((xy^\color{blue}{2})^\color{lime}{4} \rightarrow x^{\color{red}{1 \cdot \color{lime}{4}}} y^{\color{blue}{2} \cdot \color{lime}{4}} \rightarrow x^4y^8\)

\((\dfrac{x^\color{red}2}{y^\color{blue}3})^\color{lime}5 \rightarrow \dfrac{x^{\color{red}2 \cdot \color{lime}5}}{ y^{\color{blue}3 \cdot \color{lime}5}} \rightarrow \dfrac{x^{10}}{y^{15}}\)

\(\bf\small Power~ of ~Zero:\)

Anything to the power of Zero equals 1.

\(x^0 = 1\)

Examples:

\(5^0 = 1\)

\([(x^3y^5z^6)(x^{-5}y^2z^{11})]^0 = 1\)

\(\bf\small Negative~ Exponents:\)

\(\color{red}x^{-\color{blue}1} \rightarrow \dfrac{\color{blue}1}{\color{red}x}\)

\(\color{red}x^{-\color{blue}a} \rightarrow \dfrac{1}{\color{red}x^{\color{blue}a}}\)

Examples:

\(\color{red}4^{-\color{blue}1} \rightarrow \dfrac{\color{blue}1}{\color{red}4}\)

\(\color{red}3^{-\color{blue}4} \rightarrow \dfrac{1}{\color{red}3^{\color{blue}4}}\)

Answer 2

Explain it. Why does x^0=1?

Answer 3

Suppose you have \[\LARGE x^3*x^2\] is there a way we can simplify this? Well exponents just tell us how many times something is multiplied together, so we can rewrite it like that: \[\LARGE (x*x*x)*(x*x)\] but this is just 5 x's multiplied together so we get: \[\LARGE x^5\]
So we can pretty clearly see that this pattern will hold: \[\LARGE x^ax^b=x^{a+b}\]

Now let's suppose we have this fraction\[\LARGE \frac{x^3}{x^2}=\frac{x*x*x}{x*x}=x\] we just divided them all out. It looks like division follows a similar rule, we should subtract the bottom one from the top one since \[\LARGE x^{3-2}=x^1\] Now from this observation we can also see two cool observations:

if we have: \[\LARGE \frac{x^2}{x^2}=x^{2-2}=x^0\] so since anything divided by itself is 1, then by looking at these rules we just discovered we can also see that x^0=1 interesting! Also, what if there is nothing on top? Then we just have: \[\LARGE \frac{1}{x^3}=x^{0-3}=x^{-3}\] One last thing we should try to find out is what if we have something like this: \[\Large(x^2)^3=x^2*x^2*x^2=(x*x)(x*x)(x*x)=x^6\] so we can see that we just multiplied 2*3=6.

So we can figure it all out and it all has a reason for being this way as you can see by playing around with examples and testing it out.

Answer 4

Anything `except 0,` to the power of Zero equals 1