Radicals and Exponential Functions: practice solutions, Grade 9

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Practice solutions Grade 9 : Radicals and Exponential Functions — Zyro the alien explorer of Planète Maths

Written solutions to the chapter problems. Check each step, then correct yourself.

1 Perfect squares and roots ★★★

Each radicand is a perfect square.

$7^2=49$, so $\sqrt{49}=7$.
$12^2=144$, so $\sqrt{144}=12$.
$\sqrt{\dfrac{1}{9}}=\dfrac{\sqrt{1}}{\sqrt{9}}=\dfrac{1}{3}$.
$0.5^2=0.25$, so $\sqrt{0.25}=0.5$.

2 Simplify the radical ★★★

$\sqrt{20}=\sqrt{4\cdot 5}=2\sqrt{5}$.
$\sqrt{48}=\sqrt{16\cdot 3}=4\sqrt{3}$.
$\sqrt{75}=\sqrt{25\cdot 3}=5\sqrt{3}$.
$\sqrt{98}=\sqrt{49\cdot 2}=7\sqrt{2}$.

3 Add and subtract like radicals ★★★

The radicals are alike: $(4+9)\sqrt{2}=13\sqrt{2}$.
$(6-2+1)\sqrt{7}=5\sqrt{7}$.
Combine the $\sqrt{3}$ terms only: $5\sqrt{3}+2\sqrt{5}$. The last two radicals are different, so the answer cannot be shortened further.

4 Multiply radicals ★★★

$\sqrt{36}=6$.
$2\cdot 3\cdot\sqrt{5\cdot 5}=6\cdot 5=30$.
$\sqrt{60}=\sqrt{4\cdot 15}=2\sqrt{15}$.

5 Growth or decay? ★★★

Compare $b$ with 1.

$a=5$, $b=1.3>1$: growth (30% per step).
$a=200$, $b=0.9<1$: decay (10% per step).
$a=0.5$, $b=4>1$: growth.
$a=7$, $b=0.75<1$: decay (25% per step).

6 Reading a graph ★★★

$f(0)=3$, $f(1)=6$, $f(2)=12$, $f(3)=24$.
$\dfrac{6}{3}=\dfrac{12}{6}=\dfrac{24}{12}=2$, so the growth factor is $b=2$.
The initial value is 3, so $f(x)=3\cdot 2^x$.
$f(4)=3\cdot 2^4=3\cdot 16=48$.

7 Next terms ★★★

The ratio is $r=\dfrac{10}{2}=5$ (check: $\dfrac{250}{50}=5$). The next terms are $250\cdot 5=1{,}250$ and $1{,}250\cdot 5=6{,}250$.

8 Rationalize the denominator ★★★

$\dfrac{10\sqrt{5}}{5}=2\sqrt{5}$.
$\dfrac{9\sqrt{3}}{2\cdot 3}=\dfrac{9\sqrt{3}}{6}=\dfrac{3\sqrt{3}}{2}$.
$\dfrac{7\sqrt{14}}{14}=\dfrac{\sqrt{14}}{2}$.

9 A radical rectangle ★★★

$\sqrt{72}=6\sqrt{2}$ and $\sqrt{18}=3\sqrt{2}$.

Perimeter: $2(6\sqrt{2}+3\sqrt{2})=18\sqrt{2}\approx 25.46$ feet.

Area: $6\sqrt{2}\cdot 3\sqrt{2}=18\cdot 2=36$ square feet.

10 Expand with radicals ★★★

FOIL: $\sqrt{2}\cdot\sqrt{2}+5\sqrt{2}+3\sqrt{2}+15=2+8\sqrt{2}+15=17+8\sqrt{2}$.

11 A phone loses value ★★★

The decay factor is $b=1-0.20=0.8$, so $V(t)=800\cdot 0.8^t$.

$V(3)=800\cdot 0.512=\(\$409.60$\). The phone is worth $\$409.60$ after 3 years.

12 Savings account ★★★

$A=3500\cdot 1.04^5$. Since $1.04^5\approx 1.2166529$, $A\approx \(\$4{,}258.29$\).

After 5 years the account holds about $\$4{,}258.29$.

13 Linear or exponential? ★★★

Table A: the ratios $\dfrac{10}{5}=\dfrac{20}{10}=\dfrac{40}{20}=2$ are constant, so it is exponential: $y=5\cdot 2^x$.

Table B: the differences $11-7=15-11=19-15=4$ are constant, so it is linear: $y=4x+7$.

14 Find a term and its position ★★★

$a_n=6\cdot 3^{\,n-1}$.
$a_5=6\cdot 3^4=6\cdot 81=486$.
Solve $6\cdot 3^{n-1}=1458$: $3^{n-1}=243=3^5$, so $n-1=5$ and $n=6$. The sixth term is 1,458.

15 True or false? ★★★

False: the left side is $3+4=7$ and the right side is 5.
True: $\sqrt{2}\cdot\sqrt{8}=\sqrt{16}=4$.
False: the radicals are not alike, so they cannot be added. Numerically, $3\sqrt{2}+2\sqrt{3}\approx 7.71$ while $5\sqrt{5}\approx 11.18$.

16 Conjugates ★★★

Multiply by the conjugate $\sqrt{7}+1$: $\dfrac{6(\sqrt{7}+1)}{7-1}=\dfrac{6(\sqrt{7}+1)}{6}=\sqrt{7}+1$.
Multiply by $\sqrt{5}-\sqrt{3}$: $\dfrac{2(\sqrt{5}-\sqrt{3})}{5-3}=\sqrt{5}-\sqrt{3}$.

17 Squares and conjugate products ★★★

$9-12\sqrt{5}+4\cdot 5=29-12\sqrt{5}$.
The cross terms cancel: $16-3=13$.

18 A growing culture ★★★

$N(12)=40\cdot 2^{4}=40\cdot 16=640$ cells.
Solve $40\cdot 2^{t/3}=2560$: $2^{t/3}=64=2^6$, so $\dfrac{t}{3}=6$ and $t=18$. There are 2,560 cells after 18 hours.

19 Medicine in the body ★★★

After 12 hours, three half-lives have passed: $480\cdot\left(\dfrac{1}{2}\right)^3=60$ mg.
Amounts at each check: 240, 120, 60, 30, 15, 7.5 mg. At 20 hours it is 15 mg, still above 10 mg; at 24 hours it is 7.5 mg. The first check below 10 mg is at 24 hours.

20 When does exponential win? ★★★

$t=10$: $A=\(\$1{,}800$\), $B=1000\cdot 1.06^{10}\approx \(\$1{,}790.85$\), so A is ahead.

$t=11$: $A=\(\$1{,}880$\), $B\approx \(\$1{,}898.30$\), so B is ahead.

$t=12$: $A=\(\$1{,}960$\), $B\approx \(\$2{,}012.20$\).

B first exceeds A in year 11, and the gap keeps growing because B is exponential.

21 Compounding frequency ★★★

Annually: $5000\cdot 1.06^3=\(\$5{,}955.08$\).

Monthly ($n=12$, $nt=36$): $5000\left(1+\dfrac{0.06}{12}\right)^{36}=5000\cdot 1.005^{36}\approx \(\$5{,}983.40$\).

Difference: $5983.40-5955.08=\(\$28.32$\) more with monthly compounding.

22 A bouncing ball ★★★

The heights $6,\ 3.6,\ 2.16,\dots$ form a geometric sequence with ratio 0.6, so $h_n=10\cdot 0.6^n$.
$h_5=10\cdot 0.6^5=0.7776$ foot, about 9.3 inches.
$h_4=10\cdot 0.6^4=1.296$ feet (still above 1 foot) and $h_5=0.7776$ foot. So the 5th bounce is the first one under 1 foot.