Radicals and Exponential Functions: math lesson, Grade 9

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Math lessons Grade 9 : Radicals and Exponential Functions — Zyro the alien explorer of Planète Maths

Some numbers never end, like $\sqrt{2}$, and some quantities explode, like a rumor that doubles every hour. In this chapter you will learn two powerful tools. First, radicals: how to simplify them, combine them and clean up fractions that have a root in the denominator. Second, exponential functions: the equations behind growing populations, shrinking values and bank accounts that earn interest on interest.

Sommaire

1. Square roots and perfect squares

Square root

The principal square root of a number $a \ge 0$, written $\sqrt{a}$, is the nonnegative number whose square is $a$. So $\sqrt{49}=7$ because $7^2=49$. The symbol $\sqrt{\ \ }$ is called a radical and the number under it is the radicand.

A perfect square is the square of a whole number. Knowing the first twelve by heart makes every radical much easier.

n	1	2	3	4	5	6	7	8	9	10	11	12
$n^2$	1	4	9	16	25	36	49	64	81	100	121	144

When the radicand is not a perfect square, the root is irrational: its decimal never ends and never repeats. We keep it as a radical for an exact answer and use a calculator only when we need an approximation.

2. Simplifying square roots

Product rule for radicals

For $a \ge 0$ and $b \ge 0$: $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$. In the same way, $\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}$ when $b>0$.

How to simplify $\sqrt{n}$

Find the largest perfect square that divides $n$.
Write $n$ as that perfect square times another factor.
Split the root with the product rule and take the square root of the perfect square.

Example 1

Simplify $\sqrt{72}$. The largest perfect square factor is 36, since $72=36\cdot 2$. Then $\sqrt{72}=\sqrt{36}\cdot\sqrt{2}=6\sqrt{2}$.

Example 2

Simplify $\sqrt{12x^2}$ for $x\ge 0$. We have $12x^2=4\cdot 3\cdot x^2$, so $\sqrt{12x^2}=2\cdot\sqrt{3}\cdot x=2x\sqrt{3}$.

Common mistake

The root of a sum is not the sum of the roots. $\sqrt{9+16}=\sqrt{25}=5$, but $\sqrt{9}+\sqrt{16}=3+4=7$. The product rule works for multiplication only.

3. Operations with radicals

Radicals behave like variables. You can add or subtract like radicals, meaning radicals with the same radicand, by adding their coefficients: $3\sqrt{5}+4\sqrt{5}=7\sqrt{5}$. Radicals with different radicands cannot be combined, so $3\sqrt{2}+2\sqrt{3}$ stays as it is. Always simplify first, because hidden like radicals often appear.

To multiply, multiply the coefficients together and the radicands together: $a\sqrt{b}\cdot c\sqrt{d}=ac\sqrt{bd}$. To expand products of sums, use the distributive property or FOIL, exactly as with polynomials.

Example 3

Compute $\sqrt{12}+\sqrt{27}$. Simplify each root: $\sqrt{12}=2\sqrt{3}$ and $\sqrt{27}=3\sqrt{3}$. Then the sum is $2\sqrt{3}+3\sqrt{3}=5\sqrt{3}$.

Example 4

Expand $(3+\sqrt{2})^2$. Using $(a+b)^2=a^2+2ab+b^2$: $9+6\sqrt{2}+2=11+6\sqrt{2}$.

4. Rationalizing denominators

A fraction is considered fully simplified when its denominator contains no radical. To rationalize a denominator, multiply the numerator and the denominator by the same number so that the denominator becomes a whole number. The value of the fraction does not change because you are multiplying by 1.

Two cases

One radical in the denominator, such as $\dfrac{6}{\sqrt{3}}$: multiply top and bottom by $\sqrt{3}$.
A sum or difference, such as $\dfrac{4}{3+\sqrt{5}}$: multiply top and bottom by the conjugate $3-\sqrt{5}$. Since $(a+b)(a-b)=a^2-b^2$, the radical disappears.

Example 5

$\dfrac{6}{\sqrt{3}}=\dfrac{6\sqrt{3}}{\sqrt{3}\cdot\sqrt{3}}=\dfrac{6\sqrt{3}}{3}=2\sqrt{3}$.

$\dfrac{4}{3+\sqrt{5}}=\dfrac{4(3-\sqrt{5})}{(3+\sqrt{5})(3-\sqrt{5})}=\dfrac{4(3-\sqrt{5})}{9-5}=3-\sqrt{5}$.

5. Exponential growth

Exponential function

An exponential function has the form $f(x)=a\cdot b^x$, where $a\ne 0$ is the initial value (the value at $x=0$), and $b>0$, $b\ne 1$, is the growth factor. The variable is in the exponent, which is what makes it different from a polynomial.

When $b>1$ the function models exponential growth. If a quantity grows by a percent rate $r$ (written as a decimal) each period, the growth factor is $b=1+r$. A 4% increase means $b=1.04$.

Example 6

A town has 12,000 residents and grows 4% each year. The model is $P(t)=12{,}000\cdot 1.04^t$. After 3 years, $P(3)=12{,}000\cdot 1.04^3\approx 13{,}498$ residents. A bacteria colony of 200 cells that doubles every hour follows $N(t)=200\cdot 2^t$, so after 5 hours $N(5)=200\cdot 32=6{,}400$.

6. Exponential decay

When $0exponential decay. If a quantity loses a percent rate \(r$ each period, then $b=1-r$. A 15% loss gives $b=0.85$. The graph falls quickly at first, then flattens and gets closer and closer to the x-axis without ever touching it.

Example 7

A new car costs $\$24{,}000$ and loses 15% of its value every year. Its value after $t$ years is $V(t)=24{,}000\cdot 0.85^t$. After 3 years, $V(3)=24{,}000\cdot 0.614125=\(\$14{,}739$\).

7. Compound interest

When a bank adds interest to your balance and then pays interest on the new, larger balance, the interest is compounded. This is exponential growth.

Compound interest formula

\[A=P\left(1+\dfrac{r}{n}\right)^{nt}\]

Here $P$ is the principal (starting amount), $r$ the annual rate as a decimal, $n$ the number of compounding periods per year, $t$ the time in years and $A$ the final balance.

Example 8

You deposit $\$2{,}000$ at 6% compounded annually ($n=1$) for 4 years: $A=2000\cdot 1.06^4\approx \(\$2{,}524.95$\). Now take $\$1{,}500$ at 8% compounded quarterly ($n=4$) for 2 years: $A=1500\left(1+\dfrac{0.08}{4}\right)^{8}=1500\cdot 1.02^8\approx \(\$1{,}757.49$\).

The more often interest is compounded, the more you earn, but the gain from monthly instead of yearly compounding is modest. Time matters much more: look at how the bars above grow faster and faster.

8. Comparing linear and exponential models

A linear function adds the same amount each time $x$ increases by 1 (constant difference, the slope). An exponential function multiplies by the same factor each time (constant ratio). Compare $f(x)=4x+2$ and $g(x)=2^x$.

x	0	1	2	3	4	5
$f(x)=4x+2$	2	6	10	14	18	22
$g(x)=2^x$	1	2	4	8	16	32

At first the line is above the curve, but around $x\approx 4.25$ the exponential overtakes it and never looks back. Any exponential function with $a>0$ and $b>1$ eventually passes any linear function, no matter how steep the line is.

Zyro’s tip

To identify a model from a table, compute differences and ratios of consecutive outputs. Equal differences mean linear, equal ratios mean exponential. On my home planet, we check both before trusting any prediction!

9. Geometric sequences

Geometric sequence

A sequence is geometric if each term is the previous term multiplied by a constant $r$, the common ratio. The explicit formula is $a_n=a_1\cdot r^{\,n-1}$.

A geometric sequence is an exponential function whose inputs are the whole numbers $1,2,3,\dots$. To find $r$, divide any term by the one before it.

Example 9

For $3,\ 6,\ 12,\ 24,\dots$ the ratio is $r=2$, so $a_n=3\cdot 2^{n-1}$ and $a_8=3\cdot 2^7=384$. For $81,\ 27,\ 9,\dots$ the ratio is $r=\dfrac{1}{3}$, so $a_6=81\cdot\left(\dfrac{1}{3}\right)^5=\dfrac{1}{3}$.

Key takeaways

$\sqrt{ab}=\sqrt{a}\sqrt{b}$ for $a,b\ge 0$; simplify by pulling out the largest perfect square factor. Never split a sum under a root.
Add or subtract only like radicals; multiply coefficients and radicands separately.
Rationalize a denominator by multiplying by the radical itself, or by the conjugate when there is a sum or difference.
$f(x)=a\cdot b^x$: growth if $b>1$ ($b=1+r$), decay if \(0
Compound interest: $A=P\left(1+\dfrac{r}{n}\right)^{nt}$.
Linear means constant difference, exponential means constant ratio; a growing exponential always wins in the long run.
Geometric sequence: $a_n=a_1\cdot r^{\,n-1}$.