Grade 8 Algebra: Product and Quotient Rules

Name: ______________________Date: ____________Score: ____ / 6

Level 1

Same base, no coefficients

1. Simplify:
$2^3 \cdot 2^4$
2. Simplify:
$x^2 \cdot x^5$
3. Simplify:
$a \cdot a^3$
4. Simplify:
$\dfrac{5^6}{5^2}$
5. Simplify:
$\dfrac{y^9}{y^3}$
6. Simplify:
$m^4 \cdot m^4$

Answer Key

Level 1

1. $2^7$
2. $x^7$
3. $a^4$
4. $5^4$
5. $y^6$
6. $m^8$

Grade 8 Algebra: Product and Quotient Rules

Name: ______________________Date: ____________Score: ____ / 6

Level 2

With coefficients

1. Simplify:
$3x^2 \cdot 4x^3$
2. Simplify:
$(-2a^4)(5a)$
3. Simplify:
$\dfrac{12y^7}{3y^2}$
4. Simplify:
$\dfrac{20m^8}{5m^3}$
5. Simplify:
$2x^2y \cdot 3xy^4$
6. Simplify:
$\dfrac{18a^5b^3}{6a^2b}$

Answer Key

Level 2

1. $12x^5$
2. $-10a^5$
3. $4y^5$
4. $4m^5$
5. $6x^3y^5$
6. $3a^3b^2$

Grade 8 Algebra: Product and Quotient Rules

Name: ______________________Date: ____________Score: ____ / 6

Level 3

Several factors

1. Simplify:
$x^3 \cdot x^4 \cdot x^2$
2. Simplify:
$(2a^3)(3a^2)(a)$
3. Simplify:
$\dfrac{4x^5 \cdot 3x^2}{6x^3}$
4. Simplify:
$\dfrac{3^5 \cdot 3^2}{3^4}$
5. Simplify:
$\dfrac{(-3x^2y)(2xy^3)}{6x^2y^2}$
6. Simplify:
$\dfrac{10m^6n^4}{2m^2n}$

Answer Key

Level 3

1. $x^9$
2. $6a^6$
3. $2x^4$
4. $3^3$
5. $-xy^2$
6. $5m^4n^3$

Grade 8 Algebra: Power Rules

Name: ______________________Date: ____________Score: ____ / 6

Level 1

Power of a power and of a product

1. Simplify:
$(x^2)^3$
2. Simplify:
$(2^3)^2$
3. Simplify:
$(a^4)^5$
4. Simplify:
$(xy)^3$
5. Simplify:
$(3m)^2$
6. Simplify:
$\left(\dfrac{x}{y}\right)^4$

Answer Key

Level 1

1. $x^6$
2. $2^6$
3. $a^{20}$
4. $x^3y^3$
5. $9m^2$
6. $\dfrac{x^4}{y^4}$

Grade 8 Algebra: Power Rules

Name: ______________________Date: ____________Score: ____ / 6

Level 2

Coefficients and negatives

1. Simplify:
$(2x^3)^2$
2. Simplify:
$(-3a^2)^3$
3. Simplify:
$(x^2y^3)^4$
4. Simplify:
$(5m^4n)^2$
5. Simplify:
$\left(\dfrac{2a}{b^2}\right)^3$
6. Simplify:
$(-x^3)^4$

Answer Key

Level 2

1. $4x^6$
2. $-27a^6$
3. $x^8y^{12}$
4. $25m^8n^2$
5. $\dfrac{8a^3}{b^6}$
6. $x^{12}$

Grade 8 Algebra: Power Rules

Name: ______________________Date: ____________Score: ____ / 6

Level 3

Combine the rules

1. Simplify:
$(2x^2)^3 \cdot x$
2. Simplify:
$(a^2b)^3(ab^2)^2$
3. Simplify:
$\dfrac{(3x^4)^2}{9x^5}$
4. Simplify:
$(2m^3n)^2(3mn^2)$
5. Simplify:
$\dfrac{(x^3)^2 \cdot x^4}{x^5}$
6. Simplify:
$(-2y^2)^3(y^3)^2$

Answer Key

Level 3

1. $8x^7$
2. $a^8b^7$
3. $x^3$
4. $12m^7n^4$
5. $x^5$
6. $-8y^{12}$

Grade 8 Algebra: Zero and Negative Exponents

Name: ______________________Date: ____________Score: ____ / 6

Level 1

Evaluate

1. Evaluate:
$7^0$
2. Evaluate:
$(-4)^0$
3. Evaluate:
$2^{-3}$
4. Evaluate:
$5^{-1}$
5. Evaluate:
$10^{-2}$
6. Evaluate:
$3^{-2}$

Answer Key

Level 1

1. $1$
2. $1$
3. $\dfrac{1}{8}$
4. $\dfrac{1}{5}$
5. $\dfrac{1}{100}$
6. $\dfrac{1}{9}$

Grade 8 Algebra: Zero and Negative Exponents

Name: ______________________Date: ____________Score: ____ / 6

Level 2

Write with positive exponents

1. Write with positive exponents:
$x^{-4}$
2. Write with positive exponents:
$3a^{-2}$
3. Write with positive exponents:
$\dfrac{1}{y^{-5}}$
4. Write with positive exponents:
$m^{-3}n^2$
5. Evaluate:
$(2x)^0 + 3$
6. Evaluate:
$4^{-2} \cdot 4^3$

Answer Key

Level 2

1. $\dfrac{1}{x^4}$
2. $\dfrac{3}{a^2}$
3. $y^5$
4. $\dfrac{n^2}{m^3}$
5. $4$
6. $4$

Grade 8 Algebra: Zero and Negative Exponents

Name: ______________________Date: ____________Score: ____ / 6

Level 3

Simplify fully

1. Simplify:
$x^5 \cdot x^{-8}$
2. Simplify:
$\dfrac{a^{-2}}{a^4}$
3. Simplify:
$(2x^{-3})^2$
4. Simplify:
$\dfrac{12y^{-2}}{4y^{-5}}$
5. Simplify:
$(m^{-2}n^3)^{-2}$
6. Simplify:
$(3x^2y^{-1})^0 \cdot x^{-2}$

Answer Key

Level 3

1. $\dfrac{1}{x^3}$
2. $\dfrac{1}{a^6}$
3. $\dfrac{4}{x^6}$
4. $3y^3$
5. $\dfrac{m^4}{n^6}$
6. $\dfrac{1}{x^2}$

Grade 8 Algebra: Evaluating Powers

Name: ______________________Date: ____________Score: ____ / 6

Level 1

Single powers

1. Evaluate:
$2^5$
2. Evaluate:
$(-3)^2$
3. Evaluate:
$-3^2$
4. Evaluate:
$10^4$
5. Evaluate:
$\left(\tfrac{1}{2}\right)^3$
6. Evaluate:
$4^3$

Answer Key

Level 1

1. $32$
2. $9$
3. $-9$
4. $10\,000$
5. $\dfrac{1}{8}$
6. $64$

Grade 8 Algebra: Evaluating Powers

Name: ______________________Date: ____________Score: ____ / 6

Level 2

Powers with order of operations

1. Evaluate:
$2^3 + 3^2$
2. Evaluate:
$(-2)^3 \cdot 5$
3. Evaluate:
$5^2 - 4^2$
4. Evaluate:
$3 \cdot 2^4$
5. Evaluate:
$(2 + 3)^2$
6. Evaluate:
$\dfrac{6^2}{3^2}$

Answer Key

Level 2

1. $17$
2. $-40$
3. $9$
4. $48$
5. $25$
6. $4$

Grade 8 Algebra: Evaluating Powers

Name: ______________________Date: ____________Score: ____ / 6

Level 3

Variables and negative exponents

1. Evaluate for x = 3:
$x^3 - 2x$
2. Evaluate for a = -2, b = 5:
$a^2b$
3. Evaluate:
$(-1)^{25}$
4. Evaluate:
$\dfrac{(2^3)^2}{2^4}$
5. Evaluate:
$3^{-2} + 3^0$
6. Evaluate for x = 4:
$x^{-2}$

Answer Key

Level 3

1. $21$
2. $20$
3. $-1$
4. $4$
5. $\dfrac{10}{9}$
6. $\dfrac{1}{16}$

Grade 8 Algebra: Word Problems: Exponents

Name: ______________________Date: ____________Score: ____ / 6

Level 1

Repeated multiplication

1. Solve:
$\text{A}$ $\text{bacteria}$ $\text{colony}$ $\text{doubles}$ $\text{every}$ $\text{hour.}$ $\text{It}$ $\text{starts}$ $\text{with}$ $\text{1}$ $\text{cell.}$ $\text{How}$ $\text{many}$ $\text{cells}$ $\text{are}$ $\text{there}$ $\text{after}$ $\text{5}$ $\text{hours?}$
2. Solve:
$\text{Write}$ $\text{3}$ $\text{times}$ $\text{3}$ $\text{times}$ $\text{3}$ $\text{times}$ $\text{3}$ $\text{as}$ $\text{a}$ $\text{power,}$ $\text{then}$ $\text{find}$ $\text{its}$ $\text{value.}$
3. Solve:
$\text{A}$ $\text{square}$ $\text{has}$ $\text{side}$ $\text{length}$ $\text{7}$ $\text{cm.}$ $\text{Find}$ $\text{its}$ $\text{area}$ $\text{as}$ $\text{a}$ $\text{power}$ $\text{and}$ $\text{as}$ $\text{a}$ $\text{number.}$
4. Solve:
$\text{A}$ $\text{cube}$ $\text{has}$ $\text{edge}$ $\text{length}$ $\text{4}$ $\text{cm.}$ $\text{Find}$ $\text{its}$ $\text{volume}$ $\text{as}$ $\text{a}$ $\text{power}$ $\text{and}$ $\text{as}$ $\text{a}$ $\text{number.}$
5. Solve:
$\text{Ana}$ $\text{tells}$ $\text{3}$ $\text{friends}$ $\text{a}$ $\text{secret.}$ $\text{Each}$ $\text{friend}$ $\text{tells}$ $\text{3}$ $\text{new}$ $\text{people,}$ $\text{and}$ $\text{each}$ $\text{of}$ $\text{those}$ $\text{tells}$ $\text{3}$ $\text{more.}$ $\text{How}$ $\text{many}$ $\text{people}$ $\text{hear}$ $\text{it}$ $\text{in}$ $\text{the}$ $\text{third}$ $\text{round?}$
6. Solve:
$\text{Leo}$ $\text{saves}$ $\text{1}$ $\text{coin}$ $\text{on}$ $\text{day}$ $\text{1}$ $\text{and}$ $\text{doubles}$ $\text{his}$ $\text{coins}$ $\text{each}$ $\text{day.}$ $\text{How}$ $\text{many}$ $\text{coins}$ $\text{does}$ $\text{he}$ $\text{save}$ $\text{on}$ $\text{day}$ $\text{8?}$

Answer Key

Level 1

1. $2^5 = 32$
2. $3^4 = 81$
3. $7^2 = 49\text{ cm}^2$
4. $4^3 = 64\text{ cm}^3$
5. $3^3 = 27$
6. $2^7 = 128$

Grade 8 Algebra: Word Problems: Exponents

Name: ______________________Date: ____________Score: ____ / 6

Level 2

Using the exponent rules

1. Solve:
$\text{A}$ $\text{box}$ $\text{holds}$ $10^{3}\text{}$ $\text{pencils.}$ $\text{A}$ $\text{store}$ $\text{has}$ $10^{4}\text{}$ $\text{boxes.}$ $\text{How}$ $\text{many}$ $\text{pencils}$ $\text{are}$ $\text{there}$ $\text{in}$ $\text{total?}$
2. Solve:
$\text{A}$ $\text{hard}$ $\text{drive}$ $\text{holds}$ $2^{30}\text{}$ $\text{bytes.}$ $\text{Each}$ $\text{file}$ $\text{uses}$ $2^{10}\text{}$ $\text{bytes.}$ $\text{How}$ $\text{many}$ $\text{files}$ $\text{fit?}$
3. Solve:
$\text{A}$ $\text{cube}$ $\text{has}$ $\text{edge}$ $\text{length}$ $3x^{}\text{.}$ $\text{Write}$ $\text{an}$ $\text{expression}$ $\text{for}$ $\text{its}$ $\text{volume.}$
4. Solve:
$\text{A}$ $\text{square}$ $\text{garden}$ $\text{has}$ $\text{side}$ $\text{length}$ $5^{2}\text{}$ $\text{meters.}$ $\text{Find}$ $\text{its}$ $\text{area.}$
5. Solve:
$\text{A}$ $\text{rumor}$ $\text{reaches}$ $\text{4}$ $\text{times}$ $\text{as}$ $\text{many}$ $\text{people}$ $\text{each}$ $\text{day.}$ $\text{On}$ $\text{day}$ $\text{1,}$ $\text{4}$ $\text{people}$ $\text{know}$ $\text{it.}$ $\text{How}$ $\text{many}$ $\text{know}$ $\text{it}$ $\text{on}$ $\text{day}$ $\text{5?}$
6. Solve:
$\text{Light}$ $\text{travels}$ $\text{about}$ $3\times 10^{5}\text{}$ $\text{km}$ $\text{per}$ $\text{second.}$ $\text{How}$ $\text{far}$ $\text{does}$ $\text{it}$ $\text{travel}$ $\text{in}$ $10^{2}\text{}$ $\text{seconds?}$

Answer Key

Level 2

1. $10^3 \cdot 10^4 = 10^7$
2. $\dfrac{2^{30}}{2^{10}} = 2^{20}$
3. $(3x)^3 = 27x^3$
4. $(5^2)^2 = 5^4 = 625\text{ m}^2$
5. $4^5 = 1024$
6. $3 \times 10^7\text{ km}$

Grade 8 Algebra: Word Problems: Exponents

Name: ______________________Date: ____________Score: ____ / 6

Level 3

Halving, tiny numbers and comparisons

1. Solve:
$\text{A}$ $\text{ball}$ $\text{bounces}$ $\text{to}$ $\text{half}$ $\text{its}$ $\text{height}$ $\text{each}$ $\text{time.}$ $\text{It}$ $\text{is}$ $\text{dropped}$ $\text{from}$ $\text{64}$ $\text{cm.}$ $\text{How}$ $\text{high}$ $\text{does}$ $\text{it}$ $\text{bounce}$ $\text{after}$ $\text{4}$ $\text{bounces?}$
2. Solve:
$\text{A}$ $\text{cell}$ $\text{is}$ $10^{-6}\text{}$ $\text{meters}$ $\text{long.}$ $\text{How}$ $\text{many}$ $\text{cells}$ $\text{laid}$ $\text{end}$ $\text{to}$ $\text{end}$ $\text{make}$ $\text{1}$ $\text{meter?}$
3. Solve:
$\text{The}$ $\text{amount}$ $\text{of}$ $\text{medicine}$ $\text{in}$ $\text{the}$ $\text{body}$ $\text{halves}$ $\text{every}$ $\text{6}$ $\text{hours.}$ $\text{Starting}$ $\text{with}$ $\text{800}$ $\text{mg,}$ $\text{how}$ $\text{much}$ $\text{is}$ $\text{left}$ $\text{after}$ $\text{24}$ $\text{hours?}$
4. Solve:
$\text{A}$ $\text{cube}$ $\text{has}$ $\text{volume}$ $8x^{6}\text{.}$ $\text{What}$ $\text{is}$ $\text{its}$ $\text{edge}$ $\text{length?}$
5. Solve:
$\text{Which}$ $\text{is}$ $\text{greater,}$ $2^{10}\text{}$ $\text{or}$ $10^{3}\text{?}$ $\text{By}$ $\text{how}$ $\text{much?}$
6. Solve:
$\text{A}$ $\text{sheet}$ $\text{of}$ $\text{paper}$ $\text{is}$ $10^{-1}\text{}$ $\text{mm}$ $\text{thick.}$ $\text{It}$ $\text{is}$ $\text{folded}$ $\text{in}$ $\text{half}$ $\text{10}$ $\text{times.}$ $\text{How}$ $\text{thick}$ $\text{is}$ $\text{the}$ $\text{stack?}$

Answer Key

Level 3

1. $64 \cdot \left(\tfrac{1}{2}\right)^4 = 4\text{ cm}$
2. $\dfrac{1}{10^{-6}} = 10^6$
3. $800 \cdot \left(\tfrac{1}{2}\right)^4 = 50\text{ mg}$
4. $2x^2$
5. $2^{10} = 1024 > 1000,\; \text{by } 24$
6. $0.1 \cdot 2^{10} = 102.4\text{ mm}$