Exponents

Operations

In order to simply familiarize yourself with exponents, you need to understand what they are, and how they interact.

Starting with what exponents are, they are effectively a shorthand for repeated multiplication of the same value. To say \(2^5\) is the same as to say \(2 \times 2 \times 2 \times 2 \times 2\). The key difference between exponents and multiplication, however, is that exponents come first (the E is before the M in PEMDAS). At first, this does seem pretty odd. After all, exponents are really just multiplication, right? Yes, but notation really does matter. Technically multiplication is repeated addition, so why does multiplication undoubtedly come first? Mathematical convention, although not explictly proven, is important to understand and conform to in order to be successful in competition math.

Once you have a strong grasp of what an exponent does, we need to truly explore its behavior. There are no identities for adding powers. This makes sense, because how would you simplify \(2^5+2^3=2 \times 2 \times 2 \times 2 \times 2 + 2 \times 2 \times 2\), realisticall, for all powers? You can't, because you can't combine addition and multiplication. Next is multiplication and divsion. Say we have \(2^5 \times 2^3\). If we were to expand this, we would have \((2 \times 2 \times 2 \times 2 \times 2) \times (2 \times 2 \times 2)=2^8\). For all values of \(k\) and \(j\), \(x^k \times x^j=x^{k+j}\). The opposite is true for division, where we subtract. Notice that this only works for a constant base \(x\). What if we had the same exponent, but not the same base? Let's say we have \(5^2 \times 3^2\). This is \(5 \times 5 \times 3 \times 3 = 15 \times 15\). Thus, you can multiply the bases together. This is where it gets a bit complicated, because you can do something close to factoring for powers. You take the smallest power that fits into all and multiply all of the bases that's powers are raised to that number or higher and multiply them and raise it to that power, and then add in (or multiply in, is more accurate) what is left. This can help you do algebraic manipulation in some cases, but it's honestly ok to just leave it as is as well. Finally, we have powers. Let's say we have \(2^{5^3}\), I'm not goint to write this out because it would be a lot, but effectively, if you were to write this out, you would have \(3\) sets of \(2^5\), which gives \(15\) \(2\)s, for \(2^{15}\). When you raise exponents to exponents, you multiply the exponents.

In addition to these operations, there are properties of exponents that you need to know. For example, for any value \(x\), \(x^0=1\). You can check this by using the rule for division in exponents: \(\frac{x^k}{x^j}=x^{k-j}\). Setting \(k=j\), we get \(\frac{x^k}{x^k}=1=x^0\). Furthermore, exponents are invertible, but not entirely. The roots, or exponents to fractional powers (e.g. \(x^{\frac{1}{2}}=\sqrt{x}\)) in theory would reverse the operation. you can again test this by multiplying and using exponent rules. However, the caveat is that numbers raised to even powers will always return positive values, even if the input or base is negative. Thus, where \(2^2=(-2)^2\), the \(\sqrt{4}\) is denoted as \(2\), not \(-2\). One final property to consider is negative exponents. For any integer \(n\), \(x^{-n}=\frac{1}{x^n}\).

Geometric Sequences

A common application of exponents are geometric sequences, whichwhich model an exponential relationship with a common ratio. The value \(a_{n}\) of a geometric sequence can be modeled as \(a_{n}=a \times r^n\). Just as the arithmetic sequence is graphed as a linear relationship, a geometric sequence can be measured with a exponential function. A common variant on geometric sequences-unlikely to come up on the AMC 10 but good to know-is compound interest, the growth of money invested by interest. This can be modeled as \(a(1 +\frac{r}{n})^{nt}\)

Sum

The sum of a geometric series depends on whether it is said to be "convergent" or "divergent". Convergent series have \(|r| \leq 1\), meaning that oer time, they will begin to get closer and closer to \(0\). The sum for this would be \(\frac{a_{1}}{1-r}\). As you can see, there is no value for \(n\), because it should not be dependent on \(n\) because it is innfinite. A divergent, or finite series, is bounded by \(n\) and has \(|r| \geq 1\). The sum for this is \(a_{1}(\frac{1-r^n}{1-r})\)

Graphing Exponential Functions

The graph of a polynomial \(x^n\) has a rapidly increasing slope where the slope increases at a rate of change of \(n\) (calculus knowledge, not really important for the AMC 10). The similar looking yet extremely different \(2^x\) increases much more rapidly with. One key difference is that \(x^n\) will always pass through the point \((1,1)\) unless there is a phase shift, whereas \(2^x\) will pass through \((0,1)\). In general, the graph of \(a^x\) will pass through \((0,1)\) and increase more rapidly as \(a\) increases. The graph of \(x^n\) will pass through \((1,1)\) and increase more rapidly as \(n\) increases. Both graphs will be increasing for positive values of \(x\), but the exponential function will grow much faster than the polynomial function as \(x\) increases.

Conclusion

Exponents are a basic function that make up sequences and graphs. They have operations of their own and behave differently than any other function. They are a fundamental building block of mathematics and are important to understand for the AMC 10.