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MK · lesson 3 of 14 · 34 min

Exponents, Roots and Scientific Notation

Exponents and roots show up throughout Mathematics Knowledge, and the ASVAB expects you to handle them with paper and pencil only. This core-level lesson begins with what an exponent means and moves through squares and cubes, the product and quotient rules, power of a power, power of a product, zero exponents and negative exponents. A dedicated section on negative bases and parentheses explains why the placement of a minus sign matters.

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Video lesson: Exponents, Roots and Scientific Notation

Video transcript

0:00 Welcome to this explainer. If you're gearing up for the ASVAB or just want to totally master math without ever touching an calculator, you are in exactly the right place. We're going to keep this friendly, step-by-step, and completely stress-free. Here is our roadmap for today. We'll hit exponents and their rules, roots and radicals, scientific notation, some step-by-step worked examples, common traps, and finish up with a check yourself section. Okay, section one, exponents and their rules, where we'll understand the basics of powers. Exponents are really just a shorthand way to write out repeated multiplication. So, when you see a base of two with an exponent of five, it just means you multiply two by itself five times. That's two, times two, times two, times two, times two, which gives us 32. It literally just counts how many times the base is used as a factor. Now, don't let negative exponents intimidate you. Seriously, a negative exponent does not make a number negative. It just means you take the reciprocal of the base, which creates a fraction. So, ten to the power of negative three simply becomes one over ten to the third, which is one over a thousand.

1:02 See? Super easy. Roots and radicals, finding squares and cube roots. Let's smoothly transition into the exact opposite of exponents, which is finding roots. Honestly, memorizing this simple list of perfect squares from one squared is one all the way up to 15 squared is 225 is going to be your best friend. It saves you so much mental energy when you don't have a calculator. Just being able to instantly spot that the square root of 144 is 12, that's a real game changer. So, the absolutely crucial part is knowing how to use those perfect squares to simplify a messy root, like say the square root of 72. You just pull out the largest perfect square factor. Well, we know 36 goes into 72, right? So, we split it into the square root of 36 times the square root of 2. Then, take the root of 36 to get 6, leaving us with 6 times the square root of 2. Boom, done. Scientific notation and estimating, handling very large or small numbers. Scientific notation is just a super clean way to write exceptionally large or small numbers.

2:04 You simply write the number as a decimal between 1 and 10, and then multiply it by a power of 10 based on how many places you move the decimal. Right here, to turn 602,000 into 6.02, we move the decimal 5 places to the left, so our exponent is a positive 5. Step by step worked examples, applying the rules together. Now, for the fun part, we are going to apply these tools together, taking it one careful step at a time, exactly as written in your lesson. For our first example, we use the power of a power rule to turn the x to the fourth squared over on the top left into x to the eighth. You just multiply those exponents, 4 times 2 is 8. Next, we use the product rule and just add the top exponents, 8 plus 3 gives us 11. Finally, for the quotient rule, we subtract the bottom 5 from the top 11, and that smoothly leaves us with x to the sixth power. In our second example, we're looking at 5 to the negative 2. Remember that rule? The negative exponent turns the 5 into a fraction, dropping it down to the denominator to give us 1 over 5 squared. That becomes 1 over 25.

3:15 And if you think about 4 quarters, making up a whole, you know 1/25th translates exactly to 0.04. Example 3 asks us to simplify the square root of 48. Scanning our mental list of perfect squares, we know 16 goes into 48. So we pull out the perfect square of 16, take its root, which is 4, to easily get 4 times the square root of 3. For example 4, multiplying in scientific notation, we just multiply the front numbers. 5 times 7 gets us 35. Then we add the exponents on the tens. 4 plus a negative 2 gives us 2. So we've got 35 times 10 squared. But wait, we have to make a quick adjustment. Scientific notation requires the front number to be between 1 and 10. We adjust 35 down to 3.5. And because we made the front number smaller by moving the decimal left, we bump the exponent up by 1 to keep the whole thing balanced. That gives us 3.5 times 10 to the third. In our final worked example, we're dividing in scientific notation. We divide the front numbers, 6 divided by 1.5 is 4. Then we subtract the bottom exponent from the top.

4:19 So 8 minus 3 is 5. We smoothly arrive at 4 times 10 to the 5th. Look at that. Not a single calculator needed. Common traps and review. Mistakes to watch out for. We are almost at the finish line. Now everyone makes a few predictable mistakes when first learning this. So let's spotlight a very common trap so you can confidently avoid it on test day. You might be really tempted to just multiply the base by the exponent, thinking 3 times 4 is 12. No way. Do not fall for this trap. Remember, an exponent means repeated multiplication. 3 times 3 times 3 times 3 makes it 81. Keep that repeated multiplication rule front and center in your mind. Check yourself. Practice questions. We've covered all the foundational tools, so let's finish up by testing your new knowledge with two actual practice questions. Straight from the source. Question 1. What is the value of 3 to the 4th power? Is it A, 7, B, 64, C, 81 or D, 12? Take a moment to pause the explainer and think.

5:29 Okay, if you remembered our trap warning, you absolutely nailed it. The correct answer is C, 81, because the exponent tells us to use 3 as a factor 4 times. The test makers put option D right there specifically to catch people who are rushing. Let's try question 5. Re-write 4 to the negative 2 without a negative exponent. Which value is it equal to? Is it A, negative 16, B, 1/16, C, 1/8, or D, negative 8? Go ahead and pause to give it a try. The correct answer is B, 1/16. A negative exponent requires you to take the reciprocal with a positive exponent, creating a fraction 1 over 4 squared. Notice how A and D tried to trick you into making the final number negative? You didn't fall for it. This proves you now have the mental tools to tackle any of these problems absolutely calculator-free. So, now that you've mastered exponents, roots, and scientific notation by hand, what other math concepts are you going to conquer next?

Visual lesson

Lesson slides

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Lesson Goal

In this lesson you will learn to evaluate powers, apply the rules for multiplying, dividing and raising powers, and interpret zero and negative exponents. You will also find square and cube roots, recognize perfect squares, simplify radicals, and read, write and calculate with scientific notation, all without a calculator, the way the ASVAB expects.

What an Exponent Means

An exponent tells how many times a base is used as a factor. In $2^{5}$ , the base 2 appears five times:

2 \times 2 \times 2 \times 2 \times 2 = 32

It is repeated multiplication, not base times exponent, so $2^{5}$ is not 10.

Squares and Cubes

An exponent of 2 is called squared and an exponent of 3 is called cubed. For example,

9^{2} = 81 and 4^{3} = 4 \times 4 \times 4 = 64

Memorizing squares through $1 5^{2}$ and cubes through $5^{3}$ saves time on the test.

A 9 by 9 grid of 81 unit squares showing 9² = 81, and a 4 by 4 by 4 cube of small cubes showing 4³ = 64.

Product Rule for Exponents

When multiplying powers with the same base, keep the base and add the exponents:

a^{m} \cdot a^{n} = a^{m + n}

For example,

2^{3} \cdot 2^{4} = 2^{7} = 128

Three factors of 2 (2³) times four factors of 2 (2⁴) make seven factors of 2: 2⁷ = 128, since 3 + 4 = 7.

Quotient Rule for Exponents

When dividing powers with the same base, keep the base and subtract the bottom exponent from the top one:

\frac{a ^{m}}{a ^{n}} = a^{m - n}

with a not zero. For example,

\frac{k ^{9}}{k ^{4}} = k^{5}

Nine k factors over four k factors; four pairs cancel, leaving five k's: k⁹ ÷ k⁴ = k⁹⁻⁴ = k⁵.

Power of a Power

When a power is raised to another power, multiply the exponents:

(a^{m})^{n} = a^{mn}

For example,

(b^{4})^{2} = b^{8}

not $b^{6}$ .

(b⁴)² drawn as two groups of four b's, eight factors of b in all: (b⁴)² = b⁸, since 4 × 2 = 8, not 4 + 2.

Power of a Product

An exponent outside parentheses applies to every factor inside:

(ab)^{n} = a^{n} b^{n}

For example,

(3 x)^{2} = 9 x^{2} and (3 \cdot 4)^{2} = 9 \cdot 16 = 144

(3x)² as (3 · x) · (3 · x): the two 3s regroup to 3 · 3 = 9 and the two x's to x · x = x², so (3x)² = 9x².

Zero Exponents

Any nonzero number raised to the zero power equals 1:

a^{0} = 1

This follows from the quotient rule, since

\frac{a ^{3}}{a ^{3}} = a^{0}

and also equals 1. So

25 0^{0} = 1

Negative Exponents

A negative exponent means take the reciprocal of the base with a positive exponent:

a^{- n} = \frac{1}{a ^{n}}

For example,

1 0^{- 3} = \frac{1}{1000}

A negative exponent makes a fraction, never a negative number.

$Powers of ten from 10³ = 1000 down to 10⁻³ = 1/1000, each step ÷ 10; 10⁰ = 1 is the zero exponent and the negative exponents 10⁻¹, 10⁻², 10⁻³ are bracketed as fractions.$

Negative Bases and Parentheses

Parentheses show whether the minus sign belongs to the base.

(- 5)^{2} = 25

because −5 is squared, but

- 5^{2} = - 25

because only 5 is squared. Also,

(- 5)^{3} = - 125

Two panels: in (−5)² the base −5 is boxed, (−5)(−5) = 25; in −5² only the 5 is boxed, −(5 · 5) = −25.

Square Roots

A square root of a number is a value that, multiplied by itself, gives that number. The radical sign means the principal, nonnegative root, so

49 = 7

even though $(- 7)^{2}$ is also 49. Square roots of negative numbers are not real numbers, so expect only nonnegative inputs on the test.

Perfect Squares

A perfect square is the square of a whole number: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Their roots are whole numbers. Roots of other numbers fall between neighbors, such as the root of 30, which lies between 5 and 6.

Simplifying Square Roots

Split the number into a perfect-square factor times what remains, then take the root of the perfect square:

ab = a b

For example,

72 = 362 = 62

Two 6 by 6 squares of 36 make 72 = 36 × 2, so √72 = √36 × √2 = 6√2.

Cube Roots

A cube root is a number that, used as a factor three times, gives the original number. So

3125 = 5

Unlike square roots, cube roots of negative numbers are real:

3 - 64 = - 4

because

(- 4)^{3} = - 64

A 5 by 5 by 5 cube of 125 small cubes with edge 5: 5³ = 125, so ∛125 = 5.

Scientific Notation

Scientific notation writes a number as a factor from 1 up to but not including 10, times a power of ten. Large numbers get positive exponents and small numbers get negative ones. For example, 602,000 is $6.02 \times 1 0^{5}$ .

The digits of 602,000: the decimal point starts after the last 0 and hops 5 places left to a new decimal point after the 6, giving 6.02 × 10⁵.

Scientific Notation to Standard Form

To convert back, move the decimal point by the exponent: right for positive, left for negative. So $3.08 \times 1 0^{- 3}$ becomes 0.00308.

The digits 3.08 with three added zeros in front; the decimal point hops 3 places left (1, 2, 3) to give 3.08 × 10⁻³ = 0.00308.

Multiplying Scientific Notation

Multiply the front numbers and add the exponents, then adjust if the front is 10 or more. For example,

(2.5 \times 1 0^{3}) (6 \times 1 0^{4}) = 15 \times 1 0^{7} = 1.5 \times 1 0^{8}

Dividing Scientific Notation

Divide the front numbers and subtract the exponents:

\frac{8 \times 1 0 ^{9}}{2 \times 1 0 ^{3}} = 4 \times 1 0^{6}

Estimate Powers and Roots

Bracket a root between perfect squares to check answer choices. Since 81 is less than 90 and 90 is less than 100, $90$ is between 9 and 10, and closer to 9. Any choice outside that range can be eliminated at once.

Two matched number lines: 90 lies between 81 and 100, so √90 lies between 9 and 10; its dot sits just left of the 9.5 midpoint, closer to 9.

Worked Example 1 — Exponent Laws

Problem: Simplify $\frac{( x ^{4} ) ^{2} \cdot x ^{3}}{x ^{5}}$ .

Step 1: Power of a power: multiply the exponents.

(x^{4})^{2} = x^{4 \times 2} = x^{8}

Step 2: Multiplying same bases: add the exponents.

x^{8} \cdot x^{3} = x^{8 + 3} = x^{11}

Step 3: Dividing same bases: subtract the exponents.

\frac{x ^{11}}{x ^{5}} = x^{11 - 5} = x^{6}

Answer: $x^{6}$ .

Check: Count the x's. The top has

4 + 4 + 3 = 11

x's multiplied together. Canceling 5 of them with the bottom leaves 6.

Eleven x's on top in groups of 4, 4 and 3 over five x's on the bottom; five pairs cancel, leaving 6 x's: x⁶.

Worked Example 2 — Negative Exponent

Problem: Evaluate $5^{- 2}$ .

Step 1: A negative exponent means "take the reciprocal." Flip it into a fraction with a positive exponent.

5^{- 2} = \frac{1}{5 ^{2}}

Step 2: Work out the square.

5^{2} = 5 \times 5 = 25

\frac{1}{5 ^{2}} = \frac{1}{25}

Step 3: Change to a decimal. Multiply top and bottom by 4 to get hundredths.

\frac{1}{25} = \frac{4}{100} = 0.04

Answer: $\frac{1}{25}$ , or 0.04. It is positive: a negative exponent does not make the number negative.

Check:

0.04 \times 25 = 1

so 0.04 really is 1 divided by 25.

Worked Example 3 — Radical Simplification

Problem: Simplify $48$ .

Step 1: Find the largest perfect square that divides 48. Try 4, 9, 16, 25... 16 divides 48, since

16 \times 3 = 48

48 = 16 \times 3

A 4 by 12 rectangle of 48 unit squares split into three squares of 16: 48 = 16 × 3, so √48 = √16 × √3 = ?

Step 2: Split the root into two roots.

48 = 16 \times 3

Step 3: Take the root of the perfect square.

16 = 4

48 = 43

Answer: $43$ .

Check: Square the answer.

(43)^{2} = 16 \times 3 = 48

Using 4 instead of 16 (

48 = 4 \times 12

) gives $212$ , which is not fully simplified.

Worked Example 4 — Scientific Notation Product

Problem: Find $(5 \times 1 0^{4}) (7 \times 1 0^{- 2})$ .

Step 1: Multiply the front numbers.

5 \times 7 = 35

Step 2: Multiply the powers of 10 by adding the exponents.

1 0^{4} \times 1 0^{- 2} = 1 0^{4 + (- 2)}

= 1 0^{2}

Step 3: Put them together.

35 \times 1 0^{2}

Step 4: Scientific notation needs a front number from 1 to just under 10. Write 35 as 3.5 × 10, which adds 1 to the exponent.

3.5 \times 1 0^{1} \times 1 0^{2} = 3.5 \times 1 0^{3}

Answer: $3.5 \times 1 0^{3}$ , which is 3,500.

Check: In ordinary numbers, $1 0^{4}$ is 10,000 and $1 0^{- 2}$ is 0.01. So the problem is

50,000 \times 0.07 = 3,500

Worked Example 5 — Scientific Notation Quotient

Problem: Find $\frac{6 \times 1 0 ^{8}}{1.5 \times 1 0 ^{3}}$ .

Step 1: Divide the front numbers. Since

1.5 \times 4 = 6

the answer is 4.

6 \div 1.5 = 4

Step 2: Divide the powers of 10 by subtracting the exponents.

1 0^{8} \div 1 0^{3} = 1 0^{8 - 3} = 1 0^{5}

Step 3: Put them together. The front number 4 is between 1 and 10, so no adjustment is needed.

4 \times 1 0^{5}

Answer: $4 \times 1 0^{5}$ , which is 400,000.

Check: Multiply back.

4 \times 1.5 = 6 and 1 0^{5} \times 1 0^{3} = 1 0^{8}

giving the original top number.

Common Exponent and Root Traps

Multiplying the base by the exponent: $3^{4}$ is 81, not 12. Adding exponents when you should multiply them:

(a^{2})^{3} = a^{6}

not $a^{5}$ . Multiplying exponents when you should add:

x^{2} \cdot x^{3} = x^{5}

Thinking

a^{0} = 0

it equals 1. Reading a negative exponent as a negative number:

2^{- 2} = \frac{1}{4}

Leaving a front number of 10 or more in scientific notation: rewrite 12 as 1.2 and raise the exponent by one.

Quick Review

An exponent counts repeated factors. With the same base, add exponents when multiplying, subtract them when dividing, and multiply them for a power of a power. An outside exponent applies to every factor in a product. Any nonzero base to the zero power is 1, and $a^{- n}$ means $\frac{1}{a ^{n}}$ . Parentheses decide whether a negative sign is part of the base. The radical sign gives the nonnegative square root; simplify by pulling out perfect-square factors. Cube roots of negatives are negative. In scientific notation, keep the front number from 1 up to 10, multiply fronts and add exponents for products, and divide fronts and subtract exponents for quotients.

Quick review

What does an exponent mean? Is $3^{4} = 12$ ?: No. An exponent counts how many times the base is used as a factor: $3^{4} = 3 \times 3 \times 3 \times 3 = 81$ . It is repeated multiplication, not base times exponent.
Product and quotient rules for exponents?: Same base: multiply by adding exponents, $a^{m} \cdot a^{n} = a^{m + n}$ ; divide by subtracting, $\frac{a ^{m}}{a ^{n}} = a^{m - n}$ with a ≠ 0. Example: $x^{6} \cdot x^{2} = x^{8}$ and $\frac{x ^{6}}{x ^{2}} = x^{4}$ .
Power of a power and power of a product?: $(a^{m})^{n} = a^{mn}$ : multiply the exponents, so $(y^{3})^{4} = y^{12}$ . $(ab)^{n} = a^{n} b^{n}$ : the outside exponent applies to every factor, so $(2 x)^{3} = 8 x^{3}$ .
What do zero and negative exponents mean?: For nonzero a, $a^{0} = 1$ and $a^{- n} = \frac{1}{a ^{n}}$ . So $7^{0} = 1$ and $4^{- 2} = \frac{1}{16}$ . A negative exponent makes a reciprocal, never a negative number.
What does the radical sign give, and how do cube roots differ?: The radical gives the principal, nonnegative square root: $64 = 8$ , not −8; square roots of negatives are not real. Cube roots of negatives are real and negative: $3 - 27 = - 3$ .
How do you simplify a square root such as $75$ ?: Pull out the largest perfect-square factor using $ab = a b$ : $75 = 253 = 53$ . Perfect squares to know: 4, 9, 16, 25, 36, 49, 64, 81, 100, ...
What is scientific notation?: A number written as $c \times 1 0^{n}$ with $1 \leq c < 10$ . Large numbers get positive n and small numbers negative n: 48,000 = $4.8 \times 1 0^{4}$ and 0.0062 = $6.2 \times 1 0^{- 3}$ .
How do you multiply or divide in scientific notation?: Multiply (divide) the front numbers and add (subtract) the exponents, then adjust so the front is at least 1 and under 10: $(4 \times 1 0^{5}) (5 \times 1 0^{2}) = 20 \times 1 0^{7} = 2 \times 1 0^{8}$ .

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About this lesson

The second half turns to square roots, perfect squares, simplifying square roots and cube roots, then covers scientific notation: converting to standard form and multiplying or dividing numbers written that way. You will also learn to estimate powers and roots so you can eliminate unreasonable answer choices quickly.

Five worked examples cover exponent laws, a negative exponent, radical simplification and scientific notation products and quotients, followed by a list of common traps. Because MK is part of the AFQT, these skills carry weight. Budget about 34 minutes, then check yourself with the lesson test.

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