Why isn't 2^-3 equal to -8?

Because the minus sign is on the exponent, not on the base. Exponents count how many times you multiply or divide by the base; a negative exponent means divide, so 2^-3 = 1/2^3 = 1/8. Follow the pattern 8, 4, 2, 1, 1/2, 1/4, 1/8 as the exponent drops from 3 to -3 — the values keep halving and never cross zero. If you want a negative result you need a negative base or a leading minus sign, as in -2^3 = -8.

No, 0^0 is undefined in Algebra 1. The proof that a^0 = 1 relies on a^m/a^m, which requires dividing by a^m. If a = 0 that is division by zero, which is not allowed. So the zero exponent property always carries the condition that the base is not zero.

How do I know whether the exponent in scientific notation should be positive or negative?

Look at the size of the original number. If it is 10 or greater, the exponent is positive; if it is less than 1, the exponent is negative; if it is between 1 and 10, the exponent is 0. Then count how many places you moved the decimal point to get a coefficient between 1 and 10, and use that count as the size of the exponent. A quick sanity check is to convert back: multiplying by a positive power of ten should make the number bigger.

What do I do when my answer comes out as something like 34 × 10^5?

Renormalize it. Rewrite 34 as 3.4 × 10^1, so the whole thing becomes 3.4 × 10^6. Whenever you move the decimal one place left in the coefficient, add 1 to the exponent; move it one place right, subtract 1. The value stays the same because you are multiplying and dividing by the same power of ten, but the form now satisfies the requirement that the coefficient be at least 1 and less than 10.

ALG1-6.2

Zero & Negative Exponents; Scientific Notation

Learn why any nonzero base to the zero power equals 1, why negative exponents make reciprocals (not negatives), and how to convert, multiply, and divide in scientific notation.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Zero & Negative Exponents; Scientific Notation, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Zero & Negative Exponents; Scientific Notation free

What this lesson covers

You already know that

2^3

means three factors of 2. But what could

2^0

possibly mean — zero factors? And what about

2^{-3}

? Students often guess that a negative exponent produces a negative number, and that guess causes more errors in this unit than almost anything else. It does not. A negative exponent is an instruction to flip, not to change sign.

In this lesson you will build both rules from patterns you can verify yourself, so they stop feeling like arbitrary memorization. Then you will put them to work in scientific notation, the compact way scientists write the mass of an electron and the distance to a star. By the end you should be able to move a number between standard form and scientific notation in either direction, and multiply or divide two numbers written that way without reaching for a calculator.

Why $a^0 = 1$ for Every Nonzero Base

Look at a column of powers of 3 and read it from the bottom up. Each step down divides by 3.

Power	Value
$3^4$	81
$3^3$	27
$3^2$	9
$3^1$	3
$3^0$	?

Going from

3^1

3^0

means dividing 3 by 3, which gives 1. The pattern forces the answer; nobody chose it.

The quotient rule gives the same result more formally. Since

\frac{a^m}{a^m} = 1

for any nonzero

a

, and the rule for dividing like bases says

\frac{a^m}{a^m} = a^{m-m} = a^0

, the two expressions must be equal:

a^0 = 1

.

The restriction matters.

0^0

is undefined, because the argument above requires dividing by

a^m

, and you cannot divide by zero. Every other base — including negatives and fractions — gives 1:

(-7)^0 = 1

and

\left(\frac{2}{5}\right)^0 = 1

.

Where students go wrong: reading

-7^0

(-7)^0

. Without parentheses the exponent attaches only to the 7, so

-7^0 = -(7^0) = -1

. Another frequent slip is

5x^0

. Only

x

carries the exponent, so

5x^0 = 5 \cdot 1 = 5

, not 1. Read the base carefully before you apply the rule.

Negative Exponents Mean Reciprocals

Continue the powers-of-3 table past zero, still dividing by 3 at each step:

Power	Value
$3^0$	1
$3^{-1}$	$\frac{1}{3}$
$3^{-2}$	$\frac{1}{9}$
$3^{-3}$	$\frac{1}{27}$

The values shrink toward zero but never become negative. That is the whole idea, stated as a rule:

a^{-n} = \frac{1}{a^n} \quad \text{and} \quad \frac{1}{a^{-n}} = a^n

2^{-3} = \frac{1}{8}

, and

\frac{1}{4^{-2}} = 4^2 = 16

. A negative exponent moves a factor across the fraction bar and changes the sign of the exponent. It never attaches a minus sign to the value.

The most common wrong answer on a quiz is

2^{-3} = -8

. Test it against the pattern: powers of a positive base are always positive, so a negative result is impossible. A second frequent error is moving only part of a term. In

\frac{3x^{-2}}{5}

, only

x^{-2}

moves, giving

\frac{3}{5x^2}

— the 3 stays in the numerator because its exponent is positive.

For a fraction raised to a negative power, flipping the fraction is the fastest route:

\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4}

.

An expression is usually considered simplified when no negative exponents remain, so plan to rewrite them at the end of every problem.

Reading and Writing Scientific Notation

Scientific notation writes a number as

a \times 10^n

, where

1 \le |a| < 10

and

n

is an integer. The coefficient

a

carries the digits; the power of 10 records the size.

A positive exponent means a large number:

4.7 \times 10^6 = 4{,}700{,}000

. A negative exponent means a number between

-1

and 1:

3.2 \times 10^{-5} = 0.000032

. Notice again that

10^{-5}

does not make the number negative — it makes it small. Only a negative coefficient, as in

-3.2 \times 10^{5}

, makes the value negative.

To convert from scientific to standard form, move the decimal point

|n|

places: right if

n

is positive, left if

n

is negative. To go the other way, place the decimal point so exactly one nonzero digit sits to its left, then count how many places you moved it. If the original number was 10 or larger,

n

is positive; if it was smaller than 1,

n

is negative.

Standard form	Scientific notation
58,300	$5.83 \times 10^4$
0.00901	$9.01 \times 10^{-3}$
7.2	$7.2 \times 10^0$

Two things to watch. First,

47 \times 10^3

is a correct number but not correct scientific notation, because 47 is not between 1 and 10; rewrite it as

4.7 \times 10^4

. Second, count decimal places, not zeros. In 0.00901 the decimal moves three places to reach 9.01, so the exponent is

-3

Multiplying and Dividing in Scientific Notation

Because multiplication is commutative, you can regroup the coefficients together and the powers of 10 together, then use the exponent rules from the previous lesson.

(a \times 10^m)(b \times 10^n) = (ab) \times 10^{m+n}

\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}

Example of multiplying:

(3 \times 10^4)(2 \times 10^{-7}) = 6 \times 10^{-3}

. Multiply 3 and 2, then add

4 + (-7)

.

Example of dividing:

\frac{8 \times 10^5}{2 \times 10^{9}} = 4 \times 10^{-4}

. Divide 8 by 2, then subtract

5 - 9

.

The step students most often skip is renormalizing. If the new coefficient is not between 1 and 10, adjust it. Suppose you get

45 \times 10^6

. Rewrite 45 as

4.5 \times 10^1

, so the answer becomes

4.5 \times 10^7

— the exponent goes up by 1 because the coefficient shrank by a factor of 10. Going the other direction,

0.6 \times 10^{-3}

becomes

6 \times 10^{-4}

: the coefficient grew tenfold, so the exponent dropped by 1. A quick check is that moving the decimal left raises the exponent and moving it right lowers it, keeping the product unchanged.

Subtracting a negative exponent trips people up too. In

\frac{9 \times 10^{2}}{3 \times 10^{-5}}

the exponent is

2 - (-5) = 7

, giving

3 \times 10^7

. Write the subtraction out rather than doing it in your head.

Estimating and Checking Answers

Scientific notation is most useful when you use it to reason about size, not just to shorten writing. The exponent alone tells you the order of magnitude, so you can compare two quantities at a glance:

6.1 \times 10^{-9}

is smaller than

2.4 \times 10^{-8}

, even though 6.1 is bigger than 2.4, because

10^{-9}

is one tenth of

10^{-8}

. Compare exponents first; only compare coefficients when the exponents match.

That same habit makes a good error check. If a bacterium is about

2 \times 10^{-6}

meters long and a classroom is about

1 \times 10^{1}

meters long, the classroom is roughly

\frac{1 \times 10^{1}}{2 \times 10^{-6}} = 5 \times 10^{6}

times longer — about five million bacteria laid end to end. If your arithmetic had produced

5 \times 10^{-6}

, the negative exponent would immediately signal a sign error in the subtraction.

Calculators display scientific notation in a compressed form such as 5E6 or 5e6, which both mean

5 \times 10^6

. Copying that screen output literally onto homework is not acceptable notation; write it as

5 \times 10^6

.

One more useful connection: because

10^{-n} = \frac{1}{10^n}

, dividing by a power of ten is the same as multiplying by a negative power. That equivalence is what lets you convert between metric units — multiplying by

10^{-3}

turns meters into kilometers, so 2,500 meters becomes 2.5 kilometers, and the same idea powers the exponential models coming later in this unit.

Key terms

Zero exponent property.: For any nonzero number $a$ , $a^0 = 1$ . It follows from $\frac{a^m}{a^m} = a^{m-m} = a^0$ and $\frac{a^m}{a^m} = 1$ . The expression $0^0$ is undefined.
Negative exponent property.: For any nonzero $a$ and integer $n$ , $a^{-n} = \frac{1}{a^n}$ . A negative exponent indicates a reciprocal, never a negative value.
Reciprocal.: The multiplicative inverse of a number: the reciprocal of $a$ is $\frac{1}{a}$ , and their product is 1.
Scientific notation.: A number written as $a \times 10^n$ where $1 \le |a| < 10$ and $n$ is an integer.
Coefficient (in scientific notation).: The factor $a$ in $a \times 10^n$ ; it holds the significant digits and must be at least 1 and less than 10 in absolute value.
Order of magnitude.: The power of ten in a number's scientific notation, used to compare how large or small quantities are relative to one another.
Standard form (of a number).: The ordinary decimal way of writing a number, such as 4,700,000 or 0.000032.
Simplified expression.: An equivalent expression written with positive exponents only and no remaining powers that can be combined.

Worked example

Simplify

\dfrac{6.4 \times 10^{3}}{8 \times 10^{-4}}

and write the result in scientific notation. Then rewrite

\dfrac{5x^{-3}y^{0}}{2x^{2}}

with positive exponents only.

Start with the scientific notation quotient. Separate the coefficients from the powers of ten:

\frac{6.4 \times 10^{3}}{8 \times 10^{-4}} = \frac{6.4}{8} \times \frac{10^{3}}{10^{-4}}

Divide the coefficients:

6.4 \div 8 = 0.8

.

Subtract the exponents, writing the subtraction out so the double negative is visible:

3 - (-4) = 3 + 4 = 7

. So far the result is

0.8 \times 10^{7}

.

Renormalize, because 0.8 is not between 1 and 10. Moving the decimal one place right multiplies the coefficient by 10, so the exponent must drop by 1:

0.8 \times 10^7 = 8 \times 10^6

.

Check the size: dividing a few thousand by a small number near 0.0008 should give something in the millions, and

8 \times 10^6

is 8,000,000. That matches.

Now the algebraic expression. First apply the zero exponent:

y^0 = 1

, so it disappears from the numerator, leaving

\frac{5x^{-3}}{2x^{2}}

.

Combine the powers of

x

using the quotient rule:

x^{-3-2} = x^{-5}

, giving

\frac{5x^{-5}}{2}

.

Finally, rewrite the negative exponent as a reciprocal. Only the

x

factor moves; the 5 and the 2 stay where they are:

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

Practice questions

Which expression is equivalent to

4^{-2}

$-16$
$-8$
$\frac{1}{16}$
$\frac{1}{8}$

Answer: $\frac{1}{16}$

A negative exponent means take the reciprocal of the positive power, so

4^{-2} = \frac{1}{4^2} = \frac{1}{16}

. The answer

-16

comes from wrongly treating the negative sign as part of the value, and

\frac{1}{8}

comes from multiplying the base by the exponent (

4 \cdot 2

) instead of squaring the base. Powers of a positive base are always positive, so any negative choice can be ruled out immediately.

A red blood cell has a diameter of about

8 \times 10^{-6}

meters. A sheet of paper is about

1 \times 10^{-4}

meters thick. About how many red blood cells stacked side by side would match the thickness of the paper? Show your reasoning.

Answer: About 12.5 cells, since $\frac{1 \times 10^{-4}}{8 \times 10^{-6}} = 0.125 \times 10^{2} = 1.25 \times 10^{1} = 12.5$ .

Set up the division of the paper thickness by the cell diameter. Divide the coefficients:

1 \div 8 = 0.125

. Subtract the exponents:

-4 - (-6) = -4 + 6 = 2

. That gives

0.125 \times 10^{2}

, which is not yet in scientific notation because 0.125 is less than 1. Move the decimal one place right and lower the exponent by one to get

1.25 \times 10^{1}

, or 12.5 in standard form. A size check confirms it: the paper is a bit more than ten times thicker than one cell, and both quantities are tiny, so a small positive answer makes sense.

Evaluate

3x^0 + (2x)^0 + 2^{-2}

when

x = 7

, and explain why the first two terms are different.

Answer: $3 + 1 + \frac{1}{4} = 4.25$ , or $\frac{17}{4}$ .

3x^0

the exponent applies only to

x

, so

x^0 = 1

and the term equals

3 \cdot 1 = 3

. In

(2x)^0

the parentheses make the entire product the base, so the whole term equals 1. The value of

x

never matters here as long as it is not zero, which is why the substitution

x = 7

is a bit of a red herring. Finally

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

. Adding gives

3 + 1 + \frac{1}{4} = \frac{17}{4} = 4.25

. The lesson is to identify the base before applying any exponent rule.

FAQ

Why isn't $2^{-3}$ equal to $-8$ ?: Because the minus sign is on the exponent, not on the base. Exponents count how many times you multiply or divide by the base; a negative exponent means divide, so $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ . Follow the pattern $8, 4, 2, 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}$ as the exponent drops from 3 to $-3$ — the values keep halving and never cross zero. If you want a negative result you need a negative base or a leading minus sign, as in $-2^3 = -8$ .
Is $0^0$ equal to 1?: No, $0^0$ is undefined in Algebra 1. The proof that $a^0 = 1$ relies on $\frac{a^m}{a^m}$ , which requires dividing by $a^m$ . If $a = 0$ that is division by zero, which is not allowed. So the zero exponent property always carries the condition that the base is not zero.
How do I know whether the exponent in scientific notation should be positive or negative?: Look at the size of the original number. If it is 10 or greater, the exponent is positive; if it is less than 1, the exponent is negative; if it is between 1 and 10, the exponent is 0. Then count how many places you moved the decimal point to get a coefficient between 1 and 10, and use that count as the size of the exponent. A quick sanity check is to convert back: multiplying by a positive power of ten should make the number bigger.
What do I do when my answer comes out as something like $34 \times 10^{5}$ ?: Renormalize it. Rewrite 34 as $3.4 \times 10^{1}$ , so the whole thing becomes $3.4 \times 10^{6}$ . Whenever you move the decimal one place left in the coefficient, add 1 to the exponent; move it one place right, subtract 1. The value stays the same because you are multiplying and dividing by the same power of ten, but the form now satisfies the requirement that the coefficient be at least 1 and less than 10.

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