Exponents and Roots: Chapter Test

20 multiple-choice questions and 10 core practice problems, drawn from across the chapter and mixed together.

Multiple choice

20 questions, 100 points in total, 5 points each. Answer in any order and change your mind as often as you like. When you submit, your answers lock and every question shows its worked solution.

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Question 1 of 20

1

What is $4^{4}$ ?

Answer choices for question 1 A $44$ B $256$ C $16$ D $64$
2

What is $\sqrt{361}$ ?

Answer choices for question 2 A $18$ B $180.5$ C $722$ D $19$
3

Which of these is $61{,}500$ written in proper scientific notation?

Answer choices for question 3 A $6.15 \times 10^{4}$ B $61.5 \times 10^{3}$ C $6.15 \times 10^{5}$ D $6.15 \times 10^{2}$
4

Write $8^{6} \times 8^{3}$ as a single power of $8$ .

Answer choices for question 4 A $8^{18}$ B $64^{9}$ C $8^{9}$ D $8^{3}$
5

Evaluate $8^{0} + 8^{-1}$ .

Answer choices for question 5 A $\dfrac{9}{8}$ B $\dfrac{1}{8}$ C $-7$ D $9$
6

Evaluate $60 - 2 \times 5^{2}$ .

Answer choices for question 6 A $1450$ B $-40$ C $40$ D $10$
7

Which of these is $\sqrt{48}$ in simplest radical form?

Answer choices for question 7 A $2\sqrt{12}$ B $4\sqrt{3}$ C $16\sqrt{3}$ D $\sqrt{48}$ is already in simplest form
8

Which expression equals $\left(\dfrac{2}{9}\right)^{3}$ ?

Answer choices for question 8 A $\dfrac{2^{3}}{9}$ B $\dfrac{2}{9^{3}}$ C $\dfrac{2^{3}}{9^{3}}$ D $\dfrac{2 \times 3}{9 \times 3}$
9

Simplify $\dfrac{\left(5^{4}\right)^{3}}{5^{7}}$ to a single power of $5$ .

Answer choices for question 9 A $5^{19}$ B $1$ C $\dfrac{1}{5^{5}}$ D $5^{5}$
10

Compute $(2.5 \times 10^{3}) \times (6 \times 10^{4})$ , in proper scientific notation.

Answer choices for question 10 A $1.5 \times 10^{8}$ B $15 \times 10^{7}$ C $1.5 \times 10^{7}$ D $8.5 \times 10^{7}$
11

Write $\dfrac{4^{2} \times 4^{3}}{4^{8}}$ as an equivalent expression with no negative exponent.

Answer choices for question 11 A $4^{3}$ B $-4^{3}$ C $\dfrac{1}{4^{3}}$ D $4^{13}$
12

Between which two consecutive whole numbers does $\sqrt{116}$ lie?

Answer choices for question 12 A $9$ and $10$ B $10$ and $11$ C $11$ and $12$ D $57$ and $58$
13

Evaluate $(-4)^{3} + (-4)^{2}$ .

Answer choices for question 13 A $-48$ B $48$ C $-80$ D $-20$
14

Written as an ordinary number, what is $2.08 \times 10^{-5}$ ?

Answer choices for question 14 A $0.000208$ B $208{,}000$ C $-208{,}000$ D $0.0000208$
15

Exactly one of these four statements is true. Which one?

Answer choices for question 15 A $(4 + 5)^{3} = 4^{3} + 5^{3}$ B $\left(\dfrac{6}{5}\right)^{2} = \dfrac{6^{2}}{5}$ C $(4 \times 5)^{3} = 4^{3} \times 5^{3}$ D $(4 \times 5)^{3} = 4 \times 5^{3}$
16

Exactly one of these four radicals is in simplest form. Which one?

Answer choices for question 16 A $3\sqrt{20}$ B $7\sqrt{6}$ C $\sqrt{242}$ D $2\sqrt{63}$
17

Evaluate $\left(\dfrac{5}{2}\right)^{-3}$ .

Answer choices for question 17 A $\dfrac{125}{8}$ B $-\dfrac{125}{8}$ C $\dfrac{6}{5}$ D $\dfrac{8}{125}$
18

What is $\sqrt{2^{8}}$ ?

Answer choices for question 18 A $16$ B $256$ C $128$ D $4$
19

Exactly one of these four statements is true. Which one?

Answer choices for question 19 A $6 \times 10^{-3}$ is smaller than $6 \times 10^{-5}$ , because the exponent $-5$ carries the bigger digit. B $4.8 \times 10^{4}$ is larger than $52{,}000$ , because a number in scientific notation is always the more precise of the two. C $4.5 \times 10^{5}$ is larger than $390{,}000$ , because $390{,}000 = 3.9 \times 10^{5}$ and $4.5$ beats $3.9$ at the same exponent. D $3.1 \times 10^{7}$ is smaller than $9.9 \times 10^{6}$ , because $9.9$ is nearly ten while $3.1$ is barely three.
20

Exactly one of these four statements is true. Which one?

Answer choices for question 20 A $\dfrac{7^{8}}{7^{2}} = 7^{4}$ B $\left(7^{3}\right)^{2} \times 7 = 7^{7}$ C $(7 + 2)^{2} = 7^{2} + 2^{2}$ D $7^{5} + 7^{2} = 7^{7}$

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Where you stand

Evaluate a power from the definition, telling base from exponent, reading $a^{1}$ , placing a power in the order of operations, and deciding what parentheses do to a negative base. Review: Introduction to Exponents
Apply the product, quotient and power rules to powers of a single shared base, alone and in combination, reading a bare factor as a first power and knowing why the base is never altered. Review: Laws of Exponents
Spread an exponent across a product or a quotient, and say why a sum is not covered by any exponent law. Review: Laws of Exponents
Evaluate and justify zero and negative exponents, including a fraction base, keeping the reciprocal separate from the sign of the value, and carry the product, quotient and power rules through exponents that are zero or below. Review: Negative and Zero Exponents
Evaluate square roots of perfect squares, undo a square with a root, and use the principal-root convention. Review: Square Roots
Simplify a radical with the product and quotient rules for roots, run the product rule backwards, trap the root of a non-perfect square between consecutive whole numbers, and tell an exact radical form from a decimal that only approximates it. Review: Square Roots
Convert between standard form and scientific notation in both directions, respecting the range the coefficient must sit in. Review: Scientific Notation
Multiply, divide and compare in scientific notation, renormalizing the coefficient when the arithmetic pushes it out of range. Review: Scientific Notation

Core practice

10 problems from across the chapter. Work on paper, use hints when you need them, and check the answer or the full solution when you are ready.

Difficulty: Core (core-course level)

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Problem 1 of 10

Problem 1 Nines in order

Put $-9^2$ , $(-9)^2$ , $9^1$ and $(-9)^3$ in order from least to greatest.
1. Hint 1
  An exponent acts only on the base it is attached to: a whole parenthesized number, or a single number when there are no parentheses.
2. Hint 2
  Evaluate each expression before comparing, counting the negative factors, and read the exponent $1$ as one copy of its base.
Answer

$(-9)^3$ , $-9^2$ , $9^1$ , $(-9)^2$ , which are $-729$ , $-81$ , $9$ and $81$ .

Full solution

Without parentheses the exponent reaches only the $9$ , so the minus sign is applied after squaring.
$-9^2=-81$

With parentheses the base is $-9$ , and two negative factors give a positive product.
$(-9)^2=81$

A first power is a single copy of its base.
$9^1=9$

Three negative factors give a negative product.
$(-9)^3=-729$

Ordering the values $-729$ , $-81$ , $9$ and $81$ from least to greatest gives $(-9)^3$ , $-9^2$ , $9^1$ , $(-9)^2$ .

Answer

$(-9)^3$ , $-9^2$ , $9^1$ , $(-9)^2$ , which are $-729$ , $-81$ , $9$ and $81$ .

Key idea

Parentheses decide whether a minus sign belongs to the base, and a first power keeps its base.
Problem 1: Mark complete
Problem 2 A given square

Find every number whose square is $289$ , and say which of them $\sqrt{289}$ names.
1. Hint 1
  A square root undoes a square, and two opposite numbers can have the same positive square.
2. Hint 2
  Test whole numbers whose squares land near $289$ , and then consider the opposite of the one that works.
3. Hint 3
  The radical sign names the root that is zero or positive, which is called the principal root.
Answer

$17$ and $-17$ ; $\sqrt{289}=17$ .

Full solution

Test whole numbers whose squares land near $289$ .
$16^2=256$ $17^2=289$

The opposite of $17$ has the same square, since two negative factors give a positive product.
$(-17)^2=289$

Zero squares to $0$ , a positive number other than $17$ has a square below or above $289$ , and each negative number has the same square as its opposite, so $17$ and $-17$ are the only two.

The radical names the root that is zero or positive.
$\sqrt{289}=17$

Answer

$17$ and $-17$ ; $\sqrt{289}=17$ .

Key idea

A positive number has two opposite square roots, and the radical sign names the one that is not negative.
Problem 2: Mark complete
Problem 3 A recorded width

The width of a strand is recorded as $538\times10^{-11}$ m. Write the width in scientific notation and as an ordinary decimal.
1. Hint 1
  Scientific notation needs a coefficient from $1$ up to but not including $10$ , so compare the coefficient $538$ with that range first.
2. Hint 2
  Rewrite $538$ as a number in the range times a power of ten, then multiply the two powers of ten by adding their exponents.
3. Hint 3
  A negative exponent moves the decimal point to the left, by as many places as the exponent's size.
Answer

$5.38\times10^{-9}$ m, which is $0.00000000538$ m.

Full solution

The coefficient $538$ is not below $10$ , so the recorded form is not yet scientific notation.
Moving the point in $538$ two places left gives a coefficient in the range.
$538=5.38\times10^2$

Multiplying powers of ten adds the exponents, and $2+(-11)$ is $-9$ .
$10^2\times10^{-11}=10^{-9}$

So the width is $5.38\times10^{-9}$ m, and its coefficient $5.38$ lies in the range $1\le a<10$ .

A negative exponent moves the point left, here nine places, with zeros filling the empty places, which gives the ordinary decimal $0.00000000538$ m.

As a check, the first nonzero digit of $0.00000000538$ sits in the ninth decimal place, matching the exponent $-9$ .

Answer

$5.38\times10^{-9}$ m, which is $0.00000000538$ m.

Key idea

A coefficient of $10$ or more passes its extra factors of ten to the exponent, and a negative exponent then moves the decimal point to the left.
Problem 3: Mark complete
Problem 4 After the game

A stadium has $2^3$ stands. Each stand has $2^5$ rows, and each row has $2^5$ seats. After a game, $2^4$ cleaning crews share the seats equally. How many seats does each crew clean? Give the answer as a single power of $2$ .
1. Hint 1
  Count all the seats first: each layer of grouping, from stands to rows to seats, multiplies the count by another power of $2$ .
2. Hint 2
  Multiplying powers of one base adds their exponents, and sharing equally divides, which subtracts an exponent.
Answer

$2^9$ seats, which is $512$ seats.

Full solution

The seat count is the number of stands times the rows in each stand times the seats in each row.
Multiplying powers of the same base adds the exponents, and the base stays $2$ .
$2^3\times2^5\times2^5=2^{13}$

Sharing the seats equally among $2^4$ crews divides by $2^4$ , which subtracts its exponent, and $13-4$ is $9$ .
$2^{13}\div2^4=2^9$

So each crew cleans $2^9$ seats, which is $512$ seats.

As a check, there are $8\times32\times32=8192$ seats, and
$8192\div16=512.$

Answer

$2^9$ seats, which is $512$ seats.

Key idea

Equal groups inside equal groups multiply, so their exponents add, and an equal share divides, so its exponent is subtracted, with the base unchanged.
Problem 4: Mark complete
Problem 5 A difference of powers

Find the value of $(-5)^{-2}-(-5)^{-1}\times6^0$ .
1. Hint 1
  A negative exponent asks for the reciprocal of the matching positive power, and a zero power of a nonzero base is $1$ .
2. Hint 2
  The parentheses make $-5$ the whole base, so count its negative factors in each positive power before taking the reciprocal.
3. Hint 3
  Multiply before subtracting, and subtracting a negative number adds its opposite.
Answer

$\frac{6}{25}$ , or $0.24$ .

Full solution

The parentheses make $-5$ the base, and two negative factors give a positive product, so $(-5)^2=25$ .
A negative exponent asks for the reciprocal of the matching positive power.
$(-5)^{-2}=\frac{1}{25}$

One negative factor leaves the first power negative, and its reciprocal is negative too.
$(-5)^{-1}=-\frac15$

The base $6$ is nonzero, so its zero power is one.
$6^0=1$

Multiply before subtracting.
$-\frac15\times1=-\frac15$

Subtracting $-\frac15$ adds $\frac15$ , which is $\frac{5}{25}$ .
$\frac{1}{25}+\frac{5}{25}=\frac{6}{25}$

Answer

$\frac{6}{25}$ , or $0.24$ .

Key idea

A negative exponent asks for a reciprocal, and the sign of the value comes from the base and whether the exponent is odd, not from the minus sign in the exponent.
Problem 5: Mark complete
Problem 6 A combined value

Find the value of $\left(-\sqrt{144}\right)^2-\left(\sqrt{25}\right)^3$ .
1. Hint 1
  A radical returns the root that is zero or positive, and parentheses decide what a power acts on.
2. Hint 2
  Find each root first, then apply each power to everything inside its parentheses, the minus sign included.
Answer

$19$ .

Full solution

Each radical names the root that is zero or positive.
$\sqrt{144}=12$ $\sqrt{25}=5$

The parentheses make $-12$ the base of the square, and two negative factors give a positive product.
$(-12)^2=144$

The cube uses three factors of $5$ .
$5^3=125$

Subtract the cube from the square.
$144-125=19$

Answer

$19$ .

Key idea

Evaluate the root inside each base before applying its power, and let the parentheses decide whether a minus sign is part of the base.
Problem 6: Mark complete
Problem 7 Panel reports

A square panel has area $44$ square cm. Its edge is reported as $2\sqrt{11}$ cm exactly and $6.6$ cm to the nearest tenth. Are both reports correct? You may use $6.5^2=42.25$ , $6.55^2=42.9025$ , $6.6^2=43.56$ and $6.65^2=44.2225$ . Explain.
1. Hint 1
  Check the exact report by identifying a perfect-square factor of the area.
2. Hint 2
  Two of the given squares are the rounding boundaries for the reported tenth.
3. Hint 3
  Decide separately what an exact report must satisfy and what a report to the nearest tenth must satisfy.
Answer

Yes. The exact edge is $2\sqrt{11}$ cm, and it is approximately $6.6$ cm to the nearest tenth.

Full solution

The positive edge length is $\sqrt{44}$ cm.
Its radicand has a perfect-square factor.
$44=4\times11$ $\sqrt{44}=2\sqrt{11}$

The given comparisons show that $6.55^2<44$ and $44<6.65^2$ .
All three numbers are positive, so taking square roots keeps the order, and $\sqrt{44}$ lies between $6.55$ and $6.65$ .

This interval rounds to $6.6$ to the nearest tenth, giving
$\sqrt{44}\approx6.6$

Both reports are correct.
The radical is exact, while the decimal is rounded; indeed $6.6^2=43.56$ , not $44$ .

Answer

Yes. The exact edge is $2\sqrt{11}$ cm, and it is approximately $6.6$ cm to the nearest tenth.

Key idea

Bounds from squared decimal values can verify a rounded measurement while preserving its exact radical form.
Problem 7: Mark complete
Problem 8 A checked simplification

For $b\ne0$ , Kai simplifies $\frac{(b^{-4}\times b^3)^{-2}}{b^9}$ and gets $b^{-12}$ . Is Kai correct? Explain, and write the expression as a single power of $b$ .
1. Hint 1
  Work from the inside out: the product inside the parentheses comes first, then the outer power, then the division.
2. Hint 2
  A power of a power multiplies the two exponents, and the product of two negative numbers is positive.
3. Hint 3
  Once you have your own single power, test it and Kai's $b^{-12}$ at $b=2$ to see which one matches the expression.
Answer

No. The expression equals $b^{-7}$ , which is $\frac{1}{b^7}$ .

Full solution

The base is nonzero, so the laws of exponents hold for negative exponents.
Inside the parentheses, the product adds the exponents.
$b^{-4}\times b^3=b^{-1}$

A power of a power multiplies the exponents, and the product of two negative numbers is positive.
$(b^{-1})^{-2}=b^2$

Dividing by $b^9$ subtracts its exponent.
$\frac{b^2}{b^9}=b^{-7}$

So the expression equals $b^{-7}$ , which is $\frac{1}{b^7}$ , and Kai is not correct.
Kai's $b^{-12}$ is what adding $-1$ and $-2$ at the power of a power gives, $b^{-3}$ in place of $b^2$ , before the same division.

As a check at $b=2$ , the parentheses hold $\frac{1}{16}\times8$ , which is $\frac12$ , and $\left(\frac12\right)^{-2}=4$ .
$4\div2^9=\frac{1}{128}$
This matches $2^{-7}$ , while
$2^{-12}=\frac{1}{4096}.$

Answer

No. The expression equals $b^{-7}$ , which is $\frac{1}{b^7}$ .

Key idea

For a nonzero base, the laws of exponents carry through negative exponents, so a power of a power multiplies them, and a negative power raised to a negative power is a positive power.
Problem 8: Mark complete
Problem 9 Ana's equation

Ana says that $\left(\frac{3}{c}\right)^4\times c^4=3^4$ for every nonzero number $c$ . Is she correct? Explain.
1. Hint 1
  A claim about every nonzero $c$ needs a reason that works for all such values, and a single test value can only support it.
2. Hint 2
  A power of a quotient is that many copies of the fraction, so the exponent reaches both the numerator and the denominator.
3. Hint 3
  Rewrite the first factor with the exponent on its top and on its bottom, then see what multiplying by $c^4$ does to the denominator.
Answer

Yes. The expression equals $3^4$ , which is $81$ , for every nonzero $c$ .

Full solution

A power of a quotient puts the exponent on the numerator and on the denominator.
$\left(\frac{3}{c}\right)^4=\frac{3^4}{c^4}$

Since $c$ is not zero, $c^4$ is not zero either, so multiplying by $c^4$ cancels the denominator.
$\frac{3^4}{c^4}\times c^4=3^4$

The result does not depend on $c$ , so Ana is correct for every nonzero $c$ , and the value is $3^4=81$ .

As a check at $c=2$ , the first factor is $\frac{81}{16}$ , and
$\frac{81}{16}\times16=81.$

Answer

Yes. The expression equals $3^4$ , which is $81$ , for every nonzero $c$ .

Key idea

For a nonzero denominator, a power of a quotient puts the exponent on the numerator and the denominator, so multiplying by the same power of the denominator clears it.
Problem 9: Mark complete
Problem 10 A tile order

A factory packs $5\times10^3$ boxes of tiles, with $2\times10^2$ tiles in each box. Each rectangular tile is $\sqrt{13}$ cm wide and $\sqrt{52}$ cm long. The factory claims the total tile area is $2.6\times10^7$ square cm. Is the claim correct? Explain.
1. Hint 1
  Find one tile's area and the total tile count, then combine them.
2. Hint 2
  The product of the two root lengths can be written as one radical, and $52$ is $4$ times $13$ .
3. Hint 3
  Combine the coefficients and exponents in the tile count, bring that count into the standard coefficient range before multiplying by one tile's area, and check the range again at the end.
Answer

Yes. The total tile area is $2.6\times10^7$ square cm.

Full solution

Multiply the two lengths for one tile.
$\sqrt{13}\times\sqrt{52}=\sqrt{676}$
Since $676=4\times13\times13$ , the radicand is the square of $2\times13$ .
$\sqrt{676}=26$
Equivalently, $52=4\times13$ gives $\sqrt{52}=2\sqrt{13}$ , and a root times itself gives its radicand, so one tile's area $\sqrt{13}\times2\sqrt{13}$ is $2\times13$ , or $26$ .

One tile has area $26$ square cm.
The coefficient product in the total count is $5\times2=10$ , and the exponents add to $5$ .
$10\times10^5=1\times10^6$

There are $1\times10^6$ tiles, so their total area is $26\times10^6$ square cm.

The coefficient $26$ is not below $10$ , so one factor of ten moves into the power.
$26\times10^6=2.6\times10^7$

The unit is square cm, and the coefficient $2.6$ is in range, so the claimed total is correct.

Answer

Yes. The total tile area is $2.6\times10^7$ square cm.

Key idea

Combining exact root lengths with a scientific-notation count gives an exact total area, renormalized when a coefficient leaves the range.
Problem 10: Mark complete

Star problems Ten optional challenges to stretch your reasoning Advanced. This problem set goes beyond core Pre-Algebra. You can skip it. → Continue to Chapter 7: Ratios, Rates, and Percents →

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