Evaluate powers and use scientific notation

Ontario Grade 9 Mathematics

Number

Ontario Grade 9 Mathematics — Expectation B2.1

Numbers can get very large or very small very quickly. The distance from Earth to the Sun is about 150 000 000 km. The width of a human hair is about 0.00007 m. Writing and working with numbers like these is slow and easy to get wrong. Two tools — powers and scientific notation — make this manageable. This lesson builds on what you already know about multiplication and place value to show you exactly how both tools work.

What you will learn

Evaluate powers with integer bases and whole-number exponents.
Explain what a base, exponent, and power mean in plain language.
Convert large and small numbers between standard form and scientific notation.
Compare and order numbers written in scientific notation.

Reviewing the Building Blocks: What Is a Power?

You already know that multiplication is a shortcut for repeated addition:

3 + 3 + 3 + 3 = 4 \times 3

. A power is the same kind of shortcut, but for repeated multiplication instead.

Consider

3 \times 3 \times 3 \times 3

. You are multiplying 3 by itself four times. Rather than writing it out in full, you write

3^4

. Here,

3

is called the base and

4

is called the exponent. The whole expression

3^4

is called a power. The exponent tells you how many times the base is used as a factor.

Reading it aloud:

3^4

is read as '3 to the power of 4' or '3 to the 4th'. The special cases

x^2

and

x^3

are also read as 'x squared' and 'x cubed' because they match the area of a square and the volume of a cube.

3^4 = \underbrace{3 \times 3 \times 3 \times 3}_{4 \text{ factors}} = 81

A power is a shortcut for repeated multiplication.
The base is the number being multiplied repeatedly.
The exponent is the count of how many times the base appears as a factor.
Any base to the exponent $1$ equals itself: for example, $9^1 = 9$ .
Any non-zero base to the exponent $0$ equals $1$ : for example, $9^0 = 1$ .

Evaluating Powers: Positive, Negative, and Zero Exponents with Integer Bases

To evaluate a power means to calculate its value as a single number. Start by identifying the base and the exponent, then multiply the base by itself the number of times shown by the exponent.

Positive integer bases are straightforward. For example,

4^3 = 4 \times 4 \times 4 = 64

.

Negative integer bases need extra care. When the base is negative, pay close attention to brackets.

(-2)^4

means

(-2) \times (-2) \times (-2) \times (-2)

. Multiply step by step:

(-2) \times (-2) = 4

, then

4 \times (-2) = -8

, then

-8 \times (-2) = 16

. So

(-2)^4 = 16

. A negative base raised to an even exponent gives a positive result because every pair of negative factors multiplies to a positive. A negative base raised to an odd exponent gives a negative result because one negative factor is left unpaired.

Without brackets,

-2^4

means

-(2^4) = -(16) = -16

. The negative sign is not part of the base; it is applied after the power is evaluated. This is different from

(-2)^4 = 16

, so brackets matter greatly.

The zero exponent rule: any non-zero number raised to the exponent

0

equals

1

. For example,

7^0 = 1

and

(-5)^0 = 1

. You can see why by stepping down through the exponents and noticing what happens:

7^3 = 343

,

7^2 = 49

,

7^1 = 7

. Each step divides by

7

. Taking one more step — from exponent

1

down to exponent

0

— gives

7 \div 7 = 1

. So

7^0 = 1

.

(-2)^4 = (-2)\times(-2)\times(-2)\times(-2) = 16 \qquad -2^4 = -(2\times2\times2\times2) = -16

To evaluate a power, multiply the base by itself as many times as the exponent says.
A negative base inside brackets with an even exponent gives a positive result.
A negative base inside brackets with an odd exponent gives a negative result.
$-a^n$ and $(-a)^n$ are different expressions; brackets decide whether the negative sign is part of the base.
Any non-zero base raised to the exponent $0$ equals $1$ .

Scientific Notation: Writing Very Large and Very Small Numbers

Scientific notation is a standard way to write any number as a value between 1 and 10 (including 1 but not including 10), multiplied by a power of 10. In scientific notation, a number looks like

a \times 10^n

, where

a

is the coefficient and satisfies

1 \leq a < 10

, and

n

is an integer (a whole number, which can be positive, negative, or zero).

Converting a large number to scientific notation: Take 150 000 000. Move the decimal point left until exactly one non-zero digit sits to its left. Starting from 150 000 000.0, you move the decimal 8 places to the left to reach

1.5

. The number of places you moved becomes the positive exponent. So

150\,000\,000 = 1.5 \times 10^8

.

Converting a small number to scientific notation: Take 0.00007. Move the decimal point right until exactly one non-zero digit is to its left. You move it 5 places to the right to reach

7.0

. Because you moved right (meaning the original number is less than 1), the exponent is negative. So

0.00007 = 7.0 \times 10^{-5}

.

Converting back to standard form: If the exponent is positive, move the decimal point that many places to the right, filling with zeros as needed. If the exponent is negative, move the decimal point that many places to the left. For example,

3.02 \times 10^6 = 3\,020\,000

and

4.5 \times 10^{-3} = 0.0045

.

Comparing numbers in scientific notation: First compare the exponents — a larger exponent means a larger number (when both coefficients are positive and between 1 and 10). If the exponents are equal, compare the coefficients directly. For example,

3.1 \times 10^8 > 9.9 \times 10^7

because the exponent 8 is greater than 7.

150\,000\,000 = 1.5 \times 10^{8} \qquad 0.00007 = 7.0 \times 10^{-5}

Scientific notation has the form $a \times 10^n$ where $1 \leq a < 10$ and $n$ is an integer.
For numbers greater than 10, move the decimal left — the exponent is positive.
For numbers between 0 and 1, move the decimal right — the exponent is negative.
To convert back: a positive exponent means move the decimal right; a negative exponent means move the decimal left.
Compare scientific notation numbers by looking at the exponent first, then the coefficient $a$ .

Keeping Scientific Notation Proper: Checking and Adjusting

After any calculation, always check that your result is still in proper scientific notation. The coefficient must satisfy

1 \leq a < 10

. If it does not, adjust both the coefficient and the exponent together.

For example, suppose a calculation gives you

15 \times 10^4

. The coefficient 15 is not less than 10, so this is not proper scientific notation. To fix it, think about what

15

equals as a product involving a power of 10:

15 = 1.5 \times 10

, which is

1.5 \times 10^1

. So

15 \times 10^4 = 1.5 \times 10^1 \times 10^4

. Since

10^1 \times 10^4 = 10^5

(five factors of 10 in total: one from the first power and four from the second), the result is

1.5 \times 10^5

. You increased the exponent by 1 because you moved the decimal one place to the left.

Powers and scientific notation are practical tools used beyond the classroom. The mass of an electron is about

9.11 \times 10^{-31}

kg. The mass of the Sun is about

1.99 \times 10^{30}

kg. Writing either mass in standard form would take dozens of digits and be easy to miscopy. Scientific notation keeps the important digits visible and the size encoded cleanly in the exponent.

Evaluating powers of 10 accurately is what makes scientific notation reliable. When you know that

10^3 = 1000

, you immediately see that

5.6 \times 10^3 = 5600

. The two skills — evaluating powers and reading scientific notation — work together every time you use this tool.

15 \times 10^{4} = 1.5 \times 10^{5}

After any calculation, check that the coefficient satisfies $1 \leq a < 10$ .
If the coefficient is 10 or greater, move the decimal one place left and increase the exponent by 1.
If the coefficient is less than 1, move the decimal one place right and decrease the exponent by 1.
Evaluating powers of 10 is the core skill that makes scientific notation work.

Powers of 10 and Their Standard Form

Power of 10	Repeated Multiplication	Standard Form	Decimal Position
$10^5$	$10 \times 10 \times 10 \times 10 \times 10$	100 000	5 places right
$10^4$	$10 \times 10 \times 10 \times 10$	10 000	4 places right
$10^3$	$10 \times 10 \times 10$	1 000	3 places right
$10^2$	$10 \times 10$	$100$	2 places right
$10^1$	$10$	$10$	1 place right
$10^0$	(no factors of 10)	$1$	no move
$10^{-1}$	$\frac{1}{10}$	$0.1$	1 place left
$10^{-2}$	$\frac{1}{10 \times 10}$	$0.01$	2 places left

Worked example

Evaluating a Power and Expressing the Result in Scientific Notation

A science textbook states that a certain computer performs

8^5

calculations per second, and a second computer performs 32 768 calculations per second. Show that these are the same quantity, then express the number of calculations in scientific notation.

Step 1: Identify the base and exponent
The power is $8^5$ . The base is $8$ and the exponent is $5$ . This means 8 is used as a factor 5 times.
$8^5 = 8 \times 8 \times 8 \times 8 \times 8$
Step 2: Multiply from left to right, one step at a time
Work through the multiplication carefully to avoid errors. Each line multiplies the running total by 8.
$8 \times 8 = 64$ $64 \times 8 = 512$ $512 \times 8 = 4096$ $4096 \times 8 = 32\,768$
Step 3: Confirm the two values match
The textbook's value and the evaluated power are the same number, so both descriptions of the computer are equivalent.
$8^5 = 32\,768 \checkmark$
Step 4: Write 32 768 in scientific notation
Move the decimal point left until exactly one non-zero digit remains to its left. Starting from 32768.0, move left 4 places to reach 3.2768. The number of places moved is 4, so the exponent on 10 is 4.
$32\,768 = 3.2768 \times 10^{4}$
Step 5: Check that the coefficient is in the correct range
The coefficient must be at least 1 and less than 10. Here the coefficient is 3.2768, which is between 1 and 10, so the notation is correct.
$1 \leq 3.2768 < 10 \quad \checkmark$

Answer:

8^5 = 32\,768 = 3.2768 \times 10^4

Check: Convert back to standard form:

3.2768 \times 10^4

means move the decimal 4 places to the right, giving 32 768. This matches the original number, so the answer is correct.

Common mistakes and how to avoid them

Confusing

(-3)^4

and

-3^4

. Students often treat them as the same and write

81

for both.

Correction:

(-3)^4 = (-3)\times(-3)\times(-3)\times(-3) = 81

, but

-3^4 = -(3\times3\times3\times3) = -81

. Brackets decide whether the negative sign is part of the base.

Multiplying the base by the exponent instead of using repeated multiplication. For example, writing

4^3 = 12

instead of

64

.

Correction:

4^3

means

4 \times 4 \times 4

, not

4 \times 3

. The exponent counts factors; it does not multiply.

Placing the decimal in the wrong spot when converting to scientific notation. For example, writing

0.00056 = 56 \times 10^{-4}

instead of

5.6 \times 10^{-4}

.

Correction: The coefficient must satisfy

1 \leq a < 10

. Check that exactly one non-zero digit sits to the left of the decimal. If not, adjust the coefficient and the exponent together.

Using a positive exponent for a number less than 1. For example, writing

0.003 = 3 \times 10^{3}

.

Correction: Numbers less than 1 need a negative exponent.

0.003 = 3 \times 10^{-3}

. Moving the decimal to the right means the original number was small, so the exponent must be negative.

Thinking that

a^0 = 0

because 'anything times zero is zero'.

Correction:

a^0 = 1

for any non-zero base. The exponent 0 does not mean you multiply by zero. Stepping down:

5^2 = 25

,

5^1 = 5

,

5^0 = 1

. Each step divides by the base, so reaching exponent 0 gives 1.

Lesson summary

A power is shorthand for repeated multiplication: the base is the number being multiplied and the exponent tells you how many times.
To evaluate a power, multiply the base by itself as many times as the exponent indicates, working one step at a time.
A negative base inside brackets gives a positive result when the exponent is even, and a negative result when the exponent is odd; without brackets, the negative sign is applied after the power is evaluated.
Any non-zero base raised to the exponent $0$ equals $1$ .
Scientific notation expresses any number in the form $a \times 10^n$ , where $1 \leq a < 10$ and $n$ is an integer; large numbers have positive exponents and numbers between 0 and 1 have negative exponents.
To compare numbers in scientific notation, compare exponents first; if the exponents are equal, compare the coefficients.

Check your understanding

Question 1

What is the value of

(-3)^3

?

$9$
$-9$
$27$
$-27$

Show answer and explanation

-27

(-3)^3 = (-3) \times (-3) \times (-3)

. First:

(-3)\times(-3) = 9

. Then:

9 \times (-3) = -27

. A negative base with an odd exponent always gives a negative result.

Question 2

Which of the following correctly expresses

0.0000472

in scientific notation?

$472 \times 10^{-7}$
$4.72 \times 10^{-5}$
$4.72 \times 10^{5}$
$47.2 \times 10^{-6}$

Show answer and explanation

4.72 \times 10^{-5}

Move the decimal right until one non-zero digit is to its left: 0.0000472 \to 4.72, moving 5 places. Since the original number is less than 1, the exponent is negative:

4.72 \times 10^{-5}

. The coefficient

4.72

satisfies

1 \leq 4.72 < 10

.

Question 3

Which number is the largest?

$9.9 \times 10^{4}$
$1.1 \times 10^{6}$
$8.7 \times 10^{5}$
$3.5 \times 10^{5}$

Show answer and explanation

1.1 \times 10^{6}

Compare exponents first. The largest exponent is

6

, so

1.1 \times 10^6

is the largest number. In standard form: 1 100 000, which is greater than 99 000, 870 000, and 350 000.

Question 4

A student says

5^0 = 0

because 'the exponent is zero so there is nothing to multiply'. What is the correct value?

$0$ , because zero multiplications give zero.
$5$ , because you always include the base once.
$1$ , because any non-zero base to the exponent $0$ equals $1$ .
$-1$ , because the absence of multiplication makes the result negative.

Show answer and explanation

1

, because any non-zero base to the exponent

0

equals

1

.

5^0 = 1

. Step down through the exponents:

5^2 = 25

,

5^1 = 5

(divided by 5),

5^0 = 1

(divided by 5 again). Each reduction of the exponent by 1 divides by the base, so reaching exponent 0 gives 1, not 0.

Key terms

Base: The number that is multiplied repeatedly in a power. In $6^4$ , the base is $6$ .
Exponent: The small raised number that tells how many times the base is used as a factor. In $6^4$ , the exponent is $4$ .
Power: The full expression consisting of a base and an exponent, such as $6^4$ . It represents the result of repeated multiplication.
Evaluate: To calculate the single numerical value of an expression. Evaluating $6^4$ gives $1296$ .
Scientific notation: A way of writing a number as $a \times 10^n$ , where $1 \leq a < 10$ and $n$ is an integer. Example: $45\,000 = 4.5 \times 10^4$ .
Coefficient (in scientific notation): The value of $a$ in $a \times 10^n$ ; it must be at least 1 and less than 10.
Integer: Any whole number, including negative whole numbers and zero. Examples: $-3,\ 0,\ 7$ .
Standard form: A number written out in full without exponents or scientific notation. Example: 4 500 000.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation B2.1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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B2.1 · Evaluate powers and use scientific notation

Ontario Grade 9 Mathematics

Ontario Grade 9 Mathematics — Expectation B2.1

What you will learn

Reviewing the Building Blocks: What Is a Power?

Evaluating Powers: Positive, Negative, and Zero Exponents with Integer Bases

Scientific Notation: Writing Very Large and Very Small Numbers

Keeping Scientific Notation Proper: Checking and Adjusting

Powers of 10 and Their Standard Form

Evaluating a Power and Expressing the Result in Scientific Notation

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MTH1W

About this lesson and its review