Radicals and Exponential Functions: math test solutions, Grade 9 – download the PDF

1 Simplifying radicals / 4 pts

1. \(\sqrt{9\cdot 5}=3\sqrt{5}\) (1 pt)
2. \(\sqrt{64\cdot 2}=8\sqrt{2}\) (1 pt)
3. \(5\cdot 2\sqrt{3}=10\sqrt{3}\) (1 pt)
4. \(\sqrt{36}=6\) (1 pt)

2 Operations with radicals / 4 pts

1. \(\sqrt{12}=2\sqrt{3}\), so the sum is \(7\sqrt{3}-2\sqrt{3}+2\sqrt{3}=7\sqrt{3}\) (2 pts).
2. \(4-6=-2\) (1 pt)
3. \(3+8\sqrt{3}+16=19+8\sqrt{3}\) (1 pt)

3 Rationalizing / 3 pts

1. \(\dfrac{12\sqrt{6}}{6}=2\sqrt{6}\) (1 pt)
2. Multiply by the conjugate \(3+\sqrt{5}\): \(\dfrac{8(3+\sqrt{5})}{9-5}\) (1 pt) \(=2(3+\sqrt{5})=6+2\sqrt{5}\) (1 pt)

4 A growing app / 3 pts

1. The growth factor is \(1+0.15=1.15\), so \(U(t)=8000\cdot 1.15^t\) (1 pt).
2. \(U(4)=8000\cdot 1.15^4\approx 8000\cdot 1.749006=13{,}992\) users (2 pts: 1 for the substitution, 1 for the result).

5 Compound interest / 3 pts

\(A=2500\cdot 1.03^6\) (1 pt) \(\approx \(\$2{,}985.13\)\) (1 pt).

Interest earned: \(2985.13-2500=\(\$485.13\)\) (1 pt).

6 Identifying a model / 3 pts

1. The ratios \(\dfrac{12}{4}=\dfrac{36}{12}=\dfrac{108}{36}=3\) are constant, while the differences (8, 24, 72) are not. It is exponential (1 pt).
2. \(y=4\cdot 3^x\) (1 pt)
3. \(y=4\cdot 3^5=4\cdot 243=972\) (1 pt)