Exponent Rules - Pre Algebra

1. If $7^m \cdot 7^{2m} \cdot 7^{3m} = 7^{18}$, what is the value of $2m$?
   Answer: 6 Using the product rule: $7^m \cdot 7^{2m} \cdot 7^{3m} = 7^{m+2m+3m} = 7^{6m}$. Since $7^{6m} = 7^{18}$, we have $6m = 18$, so $m = 3$. Therefore, $2m = 2(3) = 6$. Choice A gives the value of $m$, not $2m$. Choice C gives the value of $3m$. Choice D gives the value of $6m$.
2. Simplify: $\frac{5^6 \cdot 5^2 \cdot 5^4}{5^3 \cdot 5^7}$. Write your answer as a single power.
   Answer: $5^2$ First simplify the numerator: $5^6 \cdot 5^2 \cdot 5^4 = 5^{6+2+4} = 5^{12}$. Then simplify the denominator: $5^3 \cdot 5^7 = 5^{3+7} = 5^{10}$. Finally apply the quotient rule: $\frac{5^{12}}{5^{10}} = 5^{12-10} = 5^2$. Choice B incorrectly adds all exponents. Choice C gives the reciprocal of the correct answer. Choice D shows an intermediate step but doesn't complete the simplification.
3. If $\frac{x^9}{x^4} = x^a$ and $x^3 \cdot x^b = x^{11}$, what is the value of $a + b$?
   Answer: 13 From the first equation: $\frac{x^9}{x^4} = x^{9-4} = x^5$, so $a = 5$. From the second equation: $x^3 \cdot x^b = x^{3+b} = x^{11}$, so $3 + b = 11$ and $b = 8$. Therefore, $a + b = 5 + 8 = 13$. Choice A gives only the value of $b$. Choice B gives the exponent from the second equation. Choice D gives $a + 9$.
4. Which of the following expressions equals $\frac{2^8}{2^3} \cdot \frac{2^6}{2^4}$? Write your answer as a single power.
   Answer: $2^7$ Simplify each fraction separately: $\frac{2^8}{2^3} = 2^{8-3} = 2^5$ and $\frac{2^6}{2^4} = 2^{6-4} = 2^2$. Then multiply: $2^5 \cdot 2^2 = 2^{5+2} = 2^7$. Choice B incorrectly adds all exponents (8+3+6+4). Choice C shows $\frac{2^{8+6}}{2^{3+4}}$ but doesn't simplify further. Choice D incorrectly multiplies exponents.
5. Simplify: $\frac{4^5 \cdot 4^3}{4^6} \cdot 4^2$
   Answer: $4^4$ First simplify the fraction: $\frac{4^5 \cdot 4^3}{4^6} = \frac{4^{5+3}}{4^6} = \frac{4^8}{4^6} = 4^{8-6} = 4^2$. Then multiply by $4^2$: $4^2 \cdot 4^2 = 4^{2+2} = 4^4$. Choice A stops after simplifying the fraction but doesn't multiply by $4^2$. Choice C incorrectly adds all exponents. Choice D shows an intermediate step without completing the calculation.
6. Which expression is equivalent to $\frac{6^4 \cdot 6^7}{6^3 \cdot 6^2} \cdot \frac{6^5}{6^8}$? Write your answer as a single power.
   Answer: $6^3$ When you encounter expressions with multiple exponents and the same base, you're working with the laws of exponents. The key is to systematically apply the rules for multiplying and dividing powers. Let's work through this step by step. First, simplify each fraction separately using the quotient rule: when dividing powers with the same base, subtract the exponents. For the first fraction: $\frac{6^4 \cdot 6^7}{6^3 \cdot 6^2}$ In the numerator, use the product rule (add exponents): $6^4 \cdot 6^7 = 6^{4+7} = 6^{11}$ In the denominator: $6^3 \cdot 6^2 = 6^{3+2} = 6^5$ So the first fraction becomes: $\frac{6^{11}}{6^5} = 6^{11-5} = 6^6$ The second fraction is already simplified: $\frac{6^5}{6^8} = 6^{5-8} = 6^{-3}$ Now multiply these results: $6^6 \cdot 6^{-3} = 6^{6+(-3)} = 6^3$ The answer is D. Let's examine why the other choices are wrong. Choice A ($\frac{6^{16}}{6^{13}}$) incorrectly adds all the exponents in numerators and denominators without proper grouping. Choice B ($6^{11}$) stops after simplifying only the first fraction. Choice C ($\frac{1}{6^3}$) makes a sign error, giving $6^{-3}$ instead of $6^3$. Remember: work systematically through exponent problems by grouping terms properly and applying one rule at a time. Don't try to do everything at once, as this leads to errors in tracking positive and negative exponents.
7. If $2^x \cdot 2^{3x} = 2^{16}$, what is the value of $x^2$?
   Answer: 16 Using the product rule for exponents, $2^x \cdot 2^{3x} = 2^{x+3x} = 2^{4x}$. Since $2^{4x} = 2^{16}$, we have $4x = 16$, so $x = 4$. Therefore, $x^2 = 4^2 = 16$. Choice A gives the value of $x$, not $x^2$. Choice B represents $2x$. Choice D represents $2^x$.
8. If $a^3 \cdot a^n = a^{11}$ and $\frac{a^{15}}{a^n} = a^7$, what is the value of $n$?
   Answer: 8 From the first equation: $a^3 \cdot a^n = a^{3+n} = a^{11}$, so $3 + n = 11$ and $n = 8$. We can verify with the second equation: $\frac{a^{15}}{a^n} = a^{15-n} = a^7$, so $15 - n = 7$ and $n = 8$. Both equations confirm $n = 8$. Choice A comes from solving only $n + 7 = 11$. Choice B comes from $15 - n = 8$. Choice D comes from misreading the first equation.
9. Which expression is equivalent to $\frac{3^8 \cdot 3^4}{3^5}$? Write your answer as a single power.
   Answer: $3^7$ First apply the product rule: $3^8 \cdot 3^4 = 3^{8+4} = 3^{12}$. Then apply the quotient rule: $\frac{3^{12}}{3^5} = 3^{12-5} = 3^7$. Choice B incorrectly adds all exponents (8+4+5). Choice C shows the intermediate step but doesn't complete the simplification. Choice D incorrectly multiplies all exponents (8×4×5).
10. If $\frac{3^{2x}}{3^x} \cdot 3^{3x} = 3^{20}$, what is the value of $x$?
    Answer: 5 First simplify the quotient: $\frac{3^{2x}}{3^x} = 3^{2x-x} = 3^x$. Then multiply: $3^x \cdot 3^{3x} = 3^{x+3x} = 3^{4x}$. Since $3^{4x} = 3^{20}$, we have $4x = 20$, so $x = 5$. Choice A comes from solving $5x = 20$. Choice C comes from solving $2x = 20$. Choice D gives the total exponent value instead of $x$.

Start this quiz