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 expressions efficiently.

The Quotient Rule for Exponents

When dividing exponential expressions with the same base, the quotient rule allows us to simplify by subtracting the exponents. For example, dividing \$4^3\( by \)4^1\( can be understood by expanding the terms: \)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\(. Therefore, the quotient rule states that for any nonzero base \)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 symbol and subtraction symbol both involve a bar, linking the operation of division with subtracting exponents. Conversely, multiplication corresponds to adding exponents.

Applying this rule to variables, if you have \)y^7\( divided by \)y^5\(, you subtract the exponents to get \)y^{7-5} = y^2\(. Similarly, dividing \)m^6\( by \)m^6\( results in \)m^{6-6} = m^0$. This introduces the 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 as repeated multiplication 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. $48$
• B. $4$
• C. $6$

Answer: (B)

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}\(. The division by \)\frac{3}{4}\( can then be viewed as a complex fraction: \)\frac{\frac{3^2}{4^2}}{\frac{3}{4}}\(.

Applying the quotient rule for exponents, which states that when dividing like bases you subtract the exponents, the expression can be separated into two parts: the numerator with base 3 and the denominator with base 4. For the base 3 terms, dividing \)3^2\( by \)3^1\( results in \)3^{2-1} = 3^1 = 3\(. Similarly, for the base 4 terms, dividing \)4^2\( by \)4^1\( results in \)4^{2-1} = 4^1 = 4\(.

Thus, the simplified form of the original expression is \)\frac{3}{4}$. This process highlights the importance of expressing numbers with common bases to effectively apply exponent rules and simplify complex fractions.

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. $-2x^3z^2$
• C. $-2x^3y^3z^2$
• D. $-2x^2z^3$

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 them fully, making calculations more efficient.

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. This rule can be understood using the quotient rule for exponents. For example, consider $m^6/m^6$. Using the quotient rule, this equals $m^0$ because $6\; -\; 6\; =\; 0$. Since any number divided by itself is one, it follows that $m^0=\; 1$ for $m\; eq\; 0$. This rule is fundamental in algebra and helps simplify expressions involving exponents.

How do the quotient rule and product rule for exponents differ?

The quotient rule and product rule for exponents are related but apply to different operations. The product rule is used when multiplying exponential expressions with the same base, and it states that you add the exponents: $a_m{a}_n=a_{m+n}$. 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 for exponents only applies when the bases of the exponential expressions are the same. If the bases differ, you cannot subtract the exponents directly. For example, $2^5/3^2$ cannot be simplified using the quotient rule because the bases 2 and 3 are different. In such cases, you must evaluate each term separately or use other algebraic methods.

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 confirm that the bases are the same, which they are (both are $y$). Then subtract the exponent in the denominator from the exponent in the numerator: $7\; -\; 5\; =\; 2$. This gives $y^2$ as the simplified expression. This method avoids expanding the powers fully and makes simplification quicker and easier.