Dividing Powers of the Same Number

Simplify $\frac{5^6}{5^2}$ and evaluate it.

Check the bases: Both powers have the base $5$ = Same base, so the quotient rule applies

Subtract the exponents (top minus bottom): $6 - 2 = 4$ = $5^4$

Evaluate the power: $5 \cdot 5 \cdot 5 \cdot 5$ = $625$

Check by calculating both powers: $5^6 \div 5^2 = 15625 \div 25$ = $625$, the same result

Answer: $\frac{5^6}{5^2} = 5^4 = 625$

Dividing Powers of a Variable

Simplify $\frac{x^9}{x^4}$ (with $x \neq 0$).

Think of the factors: $9$ factors of $x$ on top, $4$ factors of $x$ on the bottom = $4$ common factors

Cancel the common factors: Each $x$ on the bottom cancels one $x$ on top = $9 - 4 = 5$ factors stay on top

Write the result as a power: $x^{9-4}$ = $x^5$

Answer: $\frac{x^9}{x^4} = x^5$

Dividing with Coefficients

Simplify $\frac{12x^7}{4x^3}$.

Split into a number part and a power part: $\frac{12x^7}{4x^3} = \frac{12}{4} \cdot \frac{x^7}{x^3}$ = Two separate divisions

Divide the coefficients: $12 \div 4$ = $3$

Subtract the exponents: $7 - 3 = 4$ = $x^4$

Put the parts together: $3 \cdot x^4$ = $3x^4$

Answer: $\frac{12x^7}{4x^3} = 3x^4$

Two Different Variables

Simplify $\frac{18a^6b^5}{6a^2b}$.

Divide the coefficients: $18 \div 6$ = $3$

Use the rule for the base $a$: $a^{6-2}$ = $a^4$

Use the rule for the base $b$ (remember $b = b^1$): $b^{5-1}$ = $b^4$

Combine all parts: $3 \cdot a^4 \cdot b^4$ = $3a^4b^4$

Answer: $\frac{18a^6b^5}{6a^2b} = 3a^4b^4$

Product Rule First, Then Quotient Rule

Simplify $\frac{6x^5 \cdot 2x^3}{4x^6}$.

Multiply the coefficients on top: $6 \cdot 2$ = $12$

Use the product rule on top (add the exponents): $x^5 \cdot x^3 = x^{5+3}$ = Numerator: $12x^8$

Divide the coefficients: $12 \div 4$ = $3$

Use the quotient rule (subtract the exponents): $x^{8-6}$ = $x^2$

Answer: $\frac{6x^5 \cdot 2x^3}{4x^6} = 3x^2$

Mistake: Dividing the exponents: $\frac{x^8}{x^2} = x^4$

Why: Dividing the powers does not mean dividing the exponents. The $2$ factors on the bottom cancel $2$ of the $8$ factors on top, so $8 - 2$ factors stay, not $8 \div 2$.

Correct: $\frac{x^8}{x^2} = x^{8-2} = x^6$

Mistake: Subtracting in the wrong order: $\frac{3^7}{3^2} = 3^{2-7}$

Why: The exponent of the numerator always comes first. Here the top has more factors, so after cancelling the factors stay on top and the exponent is positive.

Correct: $\frac{3^7}{3^2} = 3^{7-2} = 3^5 = 243$

Mistake: Dividing the bases: $\frac{6^5}{6^2} = 1^3$

Why: The base tells you which number is multiplied, and it does not change. Only the number of factors changes.

Correct: $\frac{6^5}{6^2} = 6^{5-2} = 6^3 = 216$

Mistake: Subtracting the coefficients too: $\frac{12x^7}{4x^3} = 8x^4$

Why: Coefficients are ordinary numbers, and a fraction bar means division: $12 \div 4$. Only the exponents of the same base are subtracted.

Correct: $\frac{12x^7}{4x^3} = \frac{12}{4} \cdot x^{7-3} = 3x^4$

Photos on a Memory Card

Computer memory is counted in powers of 2, so dividing storage sizes is a job for the quotient rule.

1 MB is $2^{20}$ bytes and 1 KB is $2^{10}$ bytes, so 1 MB holds $\frac{2^{20}}{2^{10}} = 2^{10} = 1024$ KB.

How Long Sunlight Needs to Reach Earth

Astronomers divide huge distances by speeds written in scientific notation: divide the numbers in front, subtract the exponents of 10.

A distance of $6 \times 10^8$ km at $3 \times 10^5$ km per second takes $\frac{6}{3} \times 10^{8-5} = 2 \times 10^3 = 2000$ seconds.

Budget per Person

Economists divide large amounts of money by the number of people to compare countries of different sizes.

If $8 \times 10^9$ dollars are shared by $4 \times 10^6$ people, each share is $2 \times 10^3 = 2000$ dollars.

Q: Why does the rule say $a \neq 0$?

A: Because you cannot divide by zero. With $a = 0$ the denominator $0^n$ would be $0$, so the fraction would have no value at all.

Q: What happens when both exponents are equal?

A: Then every factor cancels, for example $\frac{4^3}{4^3} = 1$. The rule gives $4^{3-3} = 4^0$, and why $4^0$ equals $1$ is the topic of the lesson Zero Exponent.

Q: What if the bottom exponent is larger, like $\frac{2^3}{2^5}$?

A: Cancel as usual: $3$ factors cancel and $2$ factors of $2$ stay on the bottom, so $\frac{2^3}{2^5} = \frac{1}{2^2} = \frac{1}{4}$. How to write this with a single exponent is shown in the lesson Negative Exponents.

Q: Can I use the rule with different bases, like $\frac{6^4}{2^3}$?

A: No. The rule only counts factors of one and the same base. Calculate the powers instead: $\frac{6^4}{2^3} = \frac{1296}{8} = 162$.