The Quotient Rule: Videos & Practice Problems

The quotient rule for exponential expressions states that when dividing terms with the same base, subtract the exponent of the divisor from the exponent of the dividend: $a^m/a^n=a^{m-n}$. This simplifies expressions like $y^7/y^5=y^2$. The zero exponent rule shows any nonzero base raised to zero equals one, e.g., $m^0=\; 1$. Understanding these rules aids in simplifying polynomials and exponential terms efficiently.

The Quotient Rule for Exponents

When dividing exponential expressions with the same base, the quotient rule of exponents allows us to simplify by subtracting the exponents. For example, dividing \$4^3\( by \)4^1\( can be understood by expanding both terms: \)4^3\( is \)4 \times 4 \times 4\(, and \)4^1\( is simply \)4\(. Canceling one \)4\( from the numerator and denominator leaves \)4 \times 4\(, which is \)4^2\(. This matches the result of subtracting the exponents: \)3 - 1 = 2\(. Thus, the quotient rule states that for any nonzero base \)a\(,

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

where \)m\( and \)n\( are integers, and \)a \neq 0\(. This rule is intuitive when considering the division and subtraction symbols both involve a bar, linking the operation of division with subtracting exponents. Conversely, when multiplying exponential expressions with the same base, the exponents are added.

Applying this rule to variables, if we have \)y^7\( divided by \)y^5\(, the expression simplifies to \)y^{7-5} = y^2\(. Similarly, dividing \)m^6\( by \)m^6\( results in \)m^{6-6} = m^0$. This introduces the important zero exponent rule, which states that any nonzero number raised to the zero power equals one:

\[a^0 = 1 \quad \text{for} \quad a \neq 0\]

This can be verified by expanding both numerator and denominator and canceling identical factors, leaving the value 1. Understanding these exponent rules is essential for simplifying expressions efficiently and forms a foundation for more advanced algebraic manipulations.

Simplify the expression.
$\frac{y^{27}}{y^9}$

• A. $y^3$
• B. $y^{18}$
• C. $\frac{1}{y^{18}}$
• D. $\frac{1}{y^3}$

Answer: (B)

Simplify the expression. Hint: rewrite using exponents first.
$\frac{64}{16}$

• A. $16$
• B. $48$
• C. $4$
• D. $6$

Answer: (C)

Simplify the expression.
$(-4t{)}^0,t\ne 0$

• A. $t$
• B. $1$
• C. $0$
• D. $-1$

Answer: (B)

The Quotient Rule for Exponents Example 1

To simplify the expression nine sixteenths divided by three fourths, it is helpful to rewrite the numbers using exponential expressions with the same base. Recognizing that 9 can be expressed as 3 squared (\$3^2\() and 16 as 4 squared (\)4^2\(), the fraction \)\frac{9}{16}\( becomes \)\frac{3^2}{4^2}\(. The divisor, \)\frac{3}{4}\(, remains as is.

Applying the quotient rule for exponents, which states that when dividing like bases you subtract the exponents (\)\frac{a^m}{a^n} = a^{m-n}\(), we can rewrite the original expression as:

\[\frac{\frac{3^2}{4^2}}{\frac{3}{4}} = \frac{3^2}{3} \div \frac{4^2}{4}\]

For the numerator, dividing \)3^2\( by \)3\( is equivalent to \)3^{2-1} = 3^1 = 3\(. Similarly, for the denominator, dividing \)4^2\( by \)4\( gives \)4^{2-1} = 4^1 = 4$. Thus, the simplified expression becomes:

\[\frac{3}{4}\]>

This demonstrates how the quotient rule for exponents can be used to simplify complex fractional expressions by rewriting terms with common bases and subtracting exponents accordingly.

Use the quotient rule to rewrite each expression, then simplify.
$-\frac{12b^{11}}{4b^7}$

• A. $-3b^4$
• B. $-8b^4$
• C. $b^4$
• D. $-3b^{18}$

Answer: (A)

Use the quotient rule to rewrite each expression, then simplify.
$\frac{9x^2{y}^8}{3x{y}^2}$

• A. $6x{y}^6$
• B. $3x{y}^6$
• C. $3x^2{y}^8$
• D. $3x^2{y}^6$

Answer: (B)

Use the quotient rule to rewrite each expression, then simplify.
$\frac{30x^5{y}^3{z}^3}{-15x^2{y}^{3}z}$

• A. $2x^3{z}^2$
• B. $-2x^3{z}^2$
• C. $-2x^3{y}^3{z}^2$
• D. $-2x^2{z}^3$

Answer: (B)

Here’s what students ask on this topic:

What is the quotient rule for exponents and how is it applied?

The quotient rule for exponents states that when dividing two exponential expressions with the same base, you subtract the exponent of the denominator from the exponent of the numerator. Mathematically, this is expressed as $a_m/a_n=a_{m-n}$, where $a$ is the base and $m$ and $n$ are the exponents. For example, dividing $y^7$ by $y^5$ results in $y^2$ because $7\; -\; 5\; =\; 2$. This rule simplifies expressions by reducing the powers instead of expanding the terms fully.

Why does any nonzero number raised to the zero power equal one?

Any nonzero number raised to the zero power equals one due to the zero exponent rule, which is a direct consequence of the quotient rule. When you divide a number by itself, such as $m^6/m^6$, the result is one because all factors cancel out. Using the quotient rule, this division becomes $m^{6-6}=m^0$. Since the original expression equals one, it follows that $m^0=\; 1$ for any $m\backslash neq\; 0$. This rule is fundamental in simplifying exponential expressions and understanding powers.

How do you simplify expressions like $y^7/y^5$ using the quotient rule?

To simplify $y^7/y^5$ using the quotient rule, first identify that the base $y$ is the same in both numerator and denominator. Then subtract the exponent in the denominator from the exponent in the numerator: $7\; -\; 5\; =\; 2$. This gives $y^2$ as the simplified form. This method avoids expanding the terms fully and provides a quick way to simplify exponential expressions with the same base.

What is the difference between the product rule and the quotient rule for exponents?

The product rule and quotient rule are both used to simplify expressions with exponents but apply in different operations. The product rule is used when multiplying exponential expressions with the same base, and it states that you add the exponents: $a_m\backslash times{a}_n=a_{m+n}$. In contrast, the quotient rule applies when dividing exponential expressions with the same base, and you subtract the exponents: $a_m/a_n=a_{m-n}$. Both rules require the bases to be the same and help simplify expressions efficiently.

Can the quotient rule be applied if the bases are different?

No, the quotient rule cannot be applied if the bases are different. The quotient rule only works when dividing exponential expressions that have the same base. For example, $4^3/4^1$ can be simplified using the quotient rule by subtracting exponents, but $4^3/5^1$ cannot because the bases 4 and 5 are different. In such cases, the expressions must be handled differently, often by evaluating each term separately or using other algebraic methods.