Simplify Negative Exponents and Find Exponent Errors | Grade 8 Math | LevelKite

You already know how to find the value of a power with a negative or zero exponent, such as $2^{-3} = \frac{1}{2^{3}} = \frac{1}{8}$ and $5^{0} = 1$ . A power is a base with its exponent. In $2^{-3}$ , the base is $2$ , the number that is raised to the power, and the exponent is $-3$ .

In this lesson you use that skill in two new ways:

You simplify expressions. To simplify is to rewrite an expression as an equivalent expression, one that has the same value as the original for every allowed value of its variables. Here, a simplified expression has only positive exponents and does not repeat a base.
You check someone else's work, find the line that goes wrong, and correct it.

Every exponent in this lesson is an integer, such as $-3$ , $0$ or $4$ . Every base is a nonzero number or a variable that stands for a nonzero number. A base of zero is not allowed, because a negative exponent means dividing by the base and division by zero is undefined.

Recall what a negative exponent means for a nonzero base. A positive exponent counts how many times you multiply by the base, and a negative exponent counts how many times you divide by the base. So $3^{-2}$ means: start at $1$ and divide by $3$ two times, which gives $1 \div 3 \div 3 = \frac{1}{3 \cdot 3} = \frac{1}{3^{2}}$ .

The reciprocal of a nonzero number is $1$ divided by that number, so $3^{-2}$ is the reciprocal of $3^{2}$ . That is why you can rewrite a power with a negative exponent as $1$ over the same base with a positive exponent.

What if the negative exponent is already in a denominator, the bottom of a fraction? Then you are dividing by a fraction. Dividing by a nonzero fraction gives the same result as multiplying by its reciprocal, and the reciprocal of $\frac{1}{5^{2}}$ is $5^{2}$ . So the power moves up to the numerator, the top of the fraction:

\frac{1}{5^{-2}} = 1 \div \frac{1}{5^{2}} = 1 \times 5^{2} = 25

Key concept

A negative exponent means reciprocal

For any nonzero number $a$ and any integer $n$ , the power $a^{-n}$ is the reciprocal of $a^{n}$ . The base must be nonzero because you cannot divide by zero.

This gives you a move for simplifying. When a power of a nonzero base is a factor of a numerator or a denominator (it is multiplied by the rest, not added or subtracted), you may move it across the fraction bar if you change the sign of its exponent. Only that power moves. Every other factor stays where it is. An expression with no fraction bar can be written over a denominator of $1$ .

The negative sign in an exponent does not mean a negative value: for a nonzero base, $a^{-n}$ and $a^{n}$ always have the same sign.

a^{-n} = \frac{1}{a^{n}} \quad \text{and} \quad \frac{1}{a^{-n}} = a^{n} \qquad (a \neq 0)

Worked example

Move powers across the fraction bar

Simplify $\frac{4x^{-3}}{y^{-2}}$ , where $x$ and $y$ stand for nonzero numbers.

Find the powers with negative exponents. The power of x is a factor of the numerator, the power of y is a factor of the denominator, and x and y are nonzero. So each power can move across the fraction bar. The 4 has no negative exponent, so it stays in the numerator.
$\frac{4x^{-3}}{y^{-2}} = \frac{4 \cdot x^{-3}}{y^{-2}}$
Rewrite the power of x. Its exponent, negative 3, means dividing by x three times. Because x is nonzero, this power is the reciprocal of x to the power 3, so it moves to the denominator with the exponent 3.
$\frac{4 \cdot x^{-3}}{y^{-2}} = \frac{4}{x^{3} \cdot y^{-2}}$
Rewrite the power of y. It is in the denominator, so you are dividing by it. Because y is nonzero, dividing by a power with exponent negative 2 gives the same result as multiplying by its reciprocal, the power with exponent 2. It moves to the numerator.
$\frac{4}{x^{3} \cdot y^{-2}} = \frac{4y^{2}}{x^{3}}$
Check with x = 2 and y = 3. In the original expression, the numerator is 4 times one eighth and the denominator is one ninth. Dividing by one ninth gives the same result as multiplying by 9.
$\frac{4 \cdot \frac{1}{8}}{\frac{1}{9}} = \frac{1}{2} \div \frac{1}{9} = \frac{1}{2} \cdot 9 = \frac{9}{2}$
Find the value of the simplified form with the same numbers. The two values match. One check cannot prove that the two forms are always equal, but it can catch a mistake.
$\frac{4 \cdot 3^{2}}{2^{3}} = \frac{36}{8} = \frac{9}{2}$

Answer

For nonzero numbers $x$ and $y$ , the expression simplifies to $\frac{4y^{2}}{x^{3}}$ .

The number in front does not move. For a nonzero number $x$ , a common mistake is to rewrite $2x^{-3}$ as $\frac{1}{2x^{3}}$ . This fails because an exponent applies only to what is directly in front of it. In $2x^{-3}$ the exponent $-3$ belongs to $x$ alone, so only $x^{-3}$ becomes a reciprocal and the $2$ stays on top: $2x^{-3} = 2 \cdot \frac{1}{x^{3}} = \frac{2}{x^{3}}$ . Test $x = 2$ : the original is $2 \cdot \frac{1}{8} = \frac{1}{4}$ , but the mistaken form is $\frac{1}{2 \cdot 8} = \frac{1}{16}$ .

Parentheses change the base. In $(2x)^{-3}$ the base is $2x$ , so $(2x)^{-3} = \frac{1}{(2x)^{3}}$ . Likewise, $(-3)^{2} = 9$ has the base $-3$ , but $-3^{2} = -9$ has the base $3$ with the negative sign applied last.

You can combine powers of the same base by counting. A positive exponent counts multiplications by the base and a negative exponent counts divisions, so one multiplication and one division by the same base cancel. That gives three exponent properties, rules that hold for powers of a nonzero base with integer exponents:

Product rule. A product is the result of multiplying. Multiplying two powers of the same base puts their counts together, so you add the exponents.
Quotient rule. A quotient is the result of dividing. Dividing two powers of the same base takes away the denominator's count, so you subtract the denominator's exponent from the numerator's exponent.
Power of a power rule. Raising a power to a power multiplies the exponents. For example, $(2^{-1})^{3}$ means $2^{-1} \cdot 2^{-1} \cdot 2^{-1}$ , which is three counts of $-1$ .

If the counts cancel completely, the exponent is $0$ , and you already know that a nonzero base to the power $0$ equals $1$ . In the table, $a$ is a nonzero number and $m$ and $n$ are integers.

Exponent rules for a nonzero base and integer exponents

Rule name	Rule	Check with numbers
Product rule	$a^{m} \cdot a^{n} = a^{m+n}$ , for $a \neq 0$	$2^{5} \cdot 2^{-3} = 2^{5+(-3)} = 2^{2} = 4$ , and $32 \cdot \frac{1}{8} = 4$
Quotient rule	$\frac{a^{m}}{a^{n}} = a^{m-n}$ , for $a \neq 0$	$\frac{2^{2}}{2^{5}} = 2^{2-5} = 2^{-3} = \frac{1}{8}$ , and $\frac{4}{32} = \frac{1}{8}$
Power of a power rule	$(a^{m})^{n} = a^{m \cdot n}$ , for $a \neq 0$	$(2^{-1})^{3} = 2^{(-1) \cdot 3} = 2^{-3} = \frac{1}{8}$ , and $\left(\frac{1}{2}\right)^{3} = \frac{1}{8}$

Worked example

Combine powers, then rewrite

Simplify $\frac{6x^{-2}y^{3}}{2x^{4}y^{-1}}$ , where $x$ and $y$ stand for nonzero numbers.

Everything in the numerator and in the denominator is multiplied, so you can split the fraction into a number part, an x part and a y part. This is fraction multiplication read backward, and no denominator is zero because x and y are nonzero. The number part is 6 divided by 2, which is 3.
$\frac{6x^{-2}y^{3}}{2x^{4}y^{-1}} = \frac{6}{2} \cdot \frac{x^{-2}}{x^{4}} \cdot \frac{y^{3}}{y^{-1}}$
Use the quotient rule on the x part. It applies because both powers have the base x and x is nonzero. Subtract the denominator's exponent, 4, from the numerator's exponent, negative 2.
$\frac{x^{-2}}{x^{4}} = x^{-2-4} = x^{-6}$
Use the quotient rule on the y part, since y is nonzero. Subtracting negative 1 adds 1, because dividing by a power with exponent negative 1 undoes one division by y.
$\frac{y^{3}}{y^{-1}} = y^{3-(-1)} = y^{3+1} = y^{4}$
Multiply the three parts. The power of x still has a negative exponent. Because x is nonzero, you can move it to the denominator and change the sign of its exponent. The factors 3 and y to the power 4 stay in the numerator.
$3 \cdot x^{-6} \cdot y^{4} = \frac{3y^{4}}{x^{6}}$
Check with x = 2 and y = 2. In the original expression, the numerator is 6 times one fourth times 8, and the denominator is 2 times 16 times one half.
$\frac{6 \cdot \frac{1}{4} \cdot 8}{2 \cdot 16 \cdot \frac{1}{2}} = \frac{12}{16} = \frac{3}{4}$
Find the value of the simplified form with the same numbers. It matches, so the check finds no mistake.
$\frac{3 \cdot 2^{4}}{2^{6}} = \frac{3 \cdot 16}{64} = \frac{48}{64} = \frac{3}{4}$

Answer

For nonzero numbers $x$ and $y$ , the expression simplifies to $\frac{3y^{4}}{x^{6}}$ .

Now use the same rules to check work. Read the work one line at a time and ask of each line: are the two sides really equal? Two tools help.

Name the rule. Decide which exponent property the line uses, then compare what was written with what the rule says. Look for three things: the bases match, the right operation was done to the exponents, and every exponent kept its sign. Use the parentheses to see what each base is.
Test a value. Replace each variable with an allowed number (here, any nonzero number) and find the value on both sides of the equals sign. If the values differ, the line has an error. If they match, the line may still be wrong, so check the rule too. Do not test with $1$ : every integer power of $1$ equals $1$ , which hides exponent mistakes.

The first line that fails is where the error starts. Later lines can follow the rules and still carry the wrong result forward. The table shows four common errors, where $x$ is any nonzero number.

Four common errors (every base is nonzero)

Mistaken work	What goes wrong	Correct work
$x^{2} \cdot x^{-5} = x^{-10}$	This work multiplies the exponents. A product of powers of the same nonzero base adds them.	$x^{2} \cdot x^{-5} = x^{2+(-5)} = x^{-3} = \frac{1}{x^{3}}$
$(x^{-2})^{3} = x^{1}$	This work adds the exponents. Raising a power of a nonzero base to a power multiplies them.	$(x^{-2})^{3} = x^{(-2) \cdot 3} = x^{-6} = \frac{1}{x^{6}}$
$x^{3} \cdot x^{-3} = x^{0} = 0$	This work reads a zero exponent as the value zero. A nonzero base to the power $0$ equals $1$ .	$x^{3} \cdot x^{-3} = x^{3+(-3)} = x^{0} = 1$
$(-2)^{-4} = -\frac{1}{16}$	This work ignores the parentheses. The base is $-2$ , and four factors of $-2$ multiply to $16$ .	$(-2)^{-4} = \frac{1}{(-2)^{4}} = \frac{1}{16}$

Worked example

Find and fix an error

Maya simplified $\frac{4a^{2}}{a^{-3}}$ , where $a$ stands for a nonzero number. Line 1 of her work says $\frac{4a^{2}}{a^{-3}} = 4a^{2-3}$ . Line 2 says $4a^{2-3} = 4a^{-1}$ . Line 3 says $4a^{-1} = \frac{4}{a}$ . Find her error and correct it.

Name the rule in Line 1. Maya divides powers of the same nonzero base, so she needs the quotient rule: the numerator's exponent minus the denominator's exponent. The numerator's exponent is 2 and the denominator's exponent is negative 3. Maya subtracted 3 instead of negative 3, so the exponent lost its sign. Line 1 has an error.
$2 - (-3) = 5 \quad \text{but} \quad 2 - 3 = -1$
Confirm the error with the test value a = 2. The left side of Line 1 is 16 divided by one eighth, which is 128. The right side is 4 times one half, which is 2. The two values differ, so Line 1 is not a true statement.
$\frac{4 \cdot 2^{2}}{2^{-3}} = 16 \div \frac{1}{8} = 128 \quad \text{but} \quad 4 \cdot 2^{2-3} = 4 \cdot \frac{1}{2} = 2$
Check the other lines. Line 2 subtracts 3 from 2 correctly. Line 3 moves only the power to the denominator and keeps the 4 in the numerator. That is right, because the exponent is attached only to the variable and the variable is nonzero. Both lines follow the rules, but they carry the error from Line 1 forward.
Correct Line 1 by subtracting negative 3. The quotient rule applies because the base is nonzero. Dividing by a power with exponent negative 3 undoes three divisions by the base, so the exponent goes up by 3.
$\frac{4a^{2}}{a^{-3}} = 4a^{2-(-3)} = 4a^{2+3} = 4a^{5}$
Test the corrected result with a = 2. It gives 128, the same value as the original expression, so the test finds no mistake.
$4 \cdot 2^{5} = 4 \cdot 32 = 128$

Answer

Maya's error is in Line 1: she subtracted $3$ instead of $-3$ . For a nonzero number $a$ , the correct result is $\frac{4a^{2}}{a^{-3}} = 4a^{5}$ .

Here is the idea to keep: for a nonzero base, a negative exponent means reciprocal. So a power that is a factor can cross the fraction bar when you change the sign of its exponent, and only that power moves. Combine powers of the same nonzero base first, then rewrite any power that still has a negative exponent. When you check work, compare each line with the exponent property it uses and test a nonzero value other than $1$ . Next, you will use these properties to solve equations that have a variable in the exponent.