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

1 Perfect squares and roots ★★★

Evaluate each expression without a calculator.

1. \(\sqrt{49}\)
2. \(\sqrt{144}\)
3. \(\sqrt{\dfrac{1}{9}}\)
4. \(\sqrt{0.25}\)

2 Simplify the radical ★★★

Write each radical in simplest form.

1. \(\sqrt{20}\)
2. \(\sqrt{48}\)
3. \(\sqrt{75}\)
4. \(\sqrt{98}\)

3 Add and subtract like radicals ★★★

Simplify.

1. \(4\sqrt{2}+9\sqrt{2}\)
2. \(6\sqrt{7}-2\sqrt{7}+\sqrt{7}\)
3. \(8\sqrt{3}-3\sqrt{3}+2\sqrt{5}\)

4 Multiply radicals ★★★

Multiply and simplify.

1. \(\sqrt{3}\cdot\sqrt{12}\)
2. \(2\sqrt{5}\cdot 3\sqrt{5}\)
3. \(\sqrt{6}\cdot\sqrt{10}\)

5 Growth or decay? ★★★

For each function, give the initial value \(a\), the factor \(b\), and say whether it models growth or decay.

1. \(y=5(1.3)^x\)
2. \(y=200(0.9)^x\)
3. \(y=0.5\cdot 4^x\)
4. \(y=7\left(\dfrac{3}{4}\right)^x\)

6 Reading a graph ★★★

The graph of an exponential function \(f\) is shown below.

1. Read \(f(0)\), \(f(1)\), \(f(2)\) and \(f(3)\).
2. What is the factor between two consecutive values?
3. Write a formula for \(f(x)\).
4. Compute \(f(4)\).

7 Next terms ★★★

The sequence \(2,\ 10,\ 50,\ 250,\dots\) is geometric. Find the common ratio and the next two terms.

8 Rationalize the denominator ★★★

Rationalize each denominator.

1. \(\dfrac{10}{\sqrt{5}}\)
2. \(\dfrac{9}{2\sqrt{3}}\)
3. \(\dfrac{7}{\sqrt{14}}\)

9 A radical rectangle ★★★

A rectangular garden bed has length \(\sqrt{72}\) feet and width \(\sqrt{18}\) feet. Write the perimeter and the area in simplest radical form, then give the perimeter to the nearest hundredth.

10 Expand with radicals ★★★

Expand and simplify \((\sqrt{2}+3)(\sqrt{2}+5)\).

11 A phone loses value ★★★

A new phone costs \(\$800\) and loses 20% of its value each year. Write a function for its value after \(t\) years and find the value after 3 years.

12 Savings account ★★★

You deposit \(\$3{,}500\) in an account paying 4% interest compounded annually. How much is in the account after 5 years? Round to the nearest cent.

13 Linear or exponential? ★★★

Decide whether each table is linear or exponential and write an equation.

14 Find a term and its position ★★★

A geometric sequence has \(a_1=6\) and \(r=3\). (a) Write the explicit formula. (b) Find \(a_5\). (c) Which term equals 1,458?

15 True or false? ★★★

Say whether each statement is true or false and justify.

1. \(\sqrt{9}+\sqrt{16}=\sqrt{25}\)
2. \(\sqrt{2}\cdot\sqrt{8}=4\)
3. \(3\sqrt{2}+2\sqrt{3}=5\sqrt{5}\)

16 Conjugates ★★★

Rationalize each denominator.

1. \(\dfrac{6}{\sqrt{7}-1}\)
2. \(\dfrac{2}{\sqrt{5}+\sqrt{3}}\)

17 Squares and conjugate products ★★★

Simplify.

1. \((3-2\sqrt{5})^2\)
2. \((4+\sqrt{3})(4-\sqrt{3})\)

18 A growing culture ★★★

A lab culture starts with 40 cells and doubles every 3 hours, so \(N(t)=40\cdot 2^{t/3}\) where \(t\) is in hours. (a) How many cells are there after 12 hours? (b) After how many hours are there 2,560 cells?

19 Medicine in the body ★★★

A patient takes 480 mg of a medicine. Every 4 hours the amount in the body is cut in half. (a) How much remains after 12 hours? (b) The amount is checked every 4 hours. At the first check, how many hours after the dose is the amount below 10 mg?

20 When does exponential win? ★★★

Two accounts start with \(\$1{,}000\). Account A adds \(\$80\) each year: \(A(t)=1000+80t\). Account B earns 6% compounded annually: \(B(t)=1000\cdot 1.06^t\). Compare them at \(t=10\), \(t=11\) and \(t=12\) and decide in which year B first beats A.

21 Compounding frequency ★★★

You invest \(\$5{,}000\) at 6% for 3 years. Compare the final balance when interest is compounded annually and when it is compounded monthly. How much extra do you earn with monthly compounding? Round to the nearest cent.

22 A bouncing ball ★★★

A ball is dropped from 10 feet. After each bounce it rises to 60% of its previous height. (a) Write the height after the \(n\)th bounce. (b) What is the height after the 5th bounce? (c) After which bounce does the ball first rise to less than 1 foot?

23 Missing terms ★★★

In a geometric sequence, \(a_2=12\) and \(a_5=324\). Find the common ratio, the first term and \(a_7\).