Radical & Exponential Functions - ACT Math

1. What is $3^{-2}$?
   Answer: $\dfrac{1}{9}$ To evaluate $3^{-2}$, we apply the negative exponent rule: $a^{-n}$ = 1/$a^n$. Therefore, $3^{-2}$ = 1/$3^2$ = 1/9. The negative exponent indicates we take the reciprocal of the positive exponent result. Choice A incorrectly applies the negative sign to the base rather than using the reciprocal rule.
2. What is $11^0$?
   Answer: $1$ The zero exponent rule states that any non-zero number raised to the power of zero equals 1. 11⁰ = 1. This is a fundamental property of exponents that applies to all non-zero bases. Choice B incorrectly gives 11, confusing the base value with the result.
3. What is $3^4$?
   Answer: $81$ To evaluate $3^4$, we multiply 3 by itself four times. $3^4 = 3 \times 3 \times 3 \times 3 = 9 \times 9 = 81$. The exponential notation means repeated multiplication of the base. Choice A ($27$) would be $3^3$, not $3^4$.
4. What is $4^{3/2}$?
   Answer: $8$ To evaluate $4^{3/2}$, we can rewrite it as ($4^{1/2}$)³ or (√4)³. Since √4 = 2, we have (√4)³ = 2³ = 8. Alternatively, $4^{3/2}$ = $(2²)^{3/2}$ = $2^{2×3/2}$ = 2³ = 8. Fractional exponents represent roots and powers combined. Choice A incorrectly gives 6, which doesn't follow the fractional exponent rules.
5. Simplify: $\dfrac{\sqrt{48}}{\sqrt{3}}$
   Answer: $4$ To simplify √48/√3, we use the quotient rule for radicals: √a/√b = √(a/b). Therefore, √48/√3 = √(48/3) = √16 = 4. Since 16 is a perfect square, the radical simplifies to 4. Choice A incorrectly leaves the result as √16 instead of simplifying to 4.
6. What is $4^3$?
   Answer: $64$ To evaluate $4^3$, we multiply 4 by itself three times. $4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$. The exponential notation means repeated multiplication of the base. Choice B ($16$) would be $4^2$, not $4^3$.
7. What is the value of $x$ if $5^{x}=\dfrac{1}{25}$?
   Answer: $-2$ To solve $5^x$ = 1/25, we first express 1/25 as a power of 5. Since 25 = 5², we have 1/25 = 1/5² = $5^{-2}$. Therefore, $5^x$ = $5^{-2}$, which means x = -2. The negative exponent indicates the reciprocal relationship. Choice A incorrectly gives x = -1, which would yield $5^{-1}$ = 1/5.
8. Simplify: $\sqrt{44}$
   Answer: $2\sqrt{11}$ To simplify $\sqrt{44}$, we factor out perfect squares from under the radical. $\sqrt{44} = \sqrt{(4 \times 11)} = \sqrt{4} \times \sqrt{11} = 2\sqrt{11}$. The simplified form is $2\sqrt{11}$ since 4 is a perfect square. Choice C ($4$) would be if we incorrectly treated $\sqrt{44}$ as $\sqrt{16}$.
9. Simplify: $\sqrt{75}$
   Answer: $5\sqrt{3}$ To simplify $\sqrt{75}$, we factor out perfect squares from under the radical. $\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$. The simplified form is $5\sqrt{3}$ since 25 is a perfect square. Choice C ($15$) would be if we incorrectly treated $\sqrt{75}$ as $\sqrt{225}$.
10. Simplify: $\sqrt{32}$
    Answer: $4\sqrt{2}$ To simplify $\sqrt{32}$, we factor out perfect squares from under the radical. $\sqrt{32}$ = $\sqrt{16 \times 2}$ = $\sqrt{16} \times \sqrt{2}$ = $4\sqrt{2}$. The simplified form is $4\sqrt{2}$ since 16 is a perfect square. Choice C (8) would be if we incorrectly treated $\sqrt{32}$ as $\sqrt{64}$.
11. Simplify: $\dfrac{5^6}{5^2}$
    Answer: $5^4$ To simplify $5^6 / 5^2$, we apply the quotient rule for exponents: $a^m / a^n = a^{m-n}$. Therefore, $5^6 / 5^2 = 5^{6-2} = 5^4$. When dividing powers with the same base, we subtract the exponents. Choice A incorrectly subtracts in the wrong order ($2-6 = -4$), giving $5^3$ instead of $5^4$.
12. Simplify: $\sqrt{27y^2}$, assuming $y \ge 0$.
    Answer: $3y\sqrt{3}$ To simplify $\sqrt{27y^2}$, we factor $27 = 9 \times 3$ where 9 is a perfect square. Since $y \ge 0$, we have $\sqrt{y^2} = y$. Therefore, $\sqrt{27y^2} = \sqrt{9 \times 3 \times y^2} = \sqrt{9} \times \sqrt{3} \times \sqrt{y^2} = 3 \times \sqrt{3} \times y = 3y\sqrt{3}$. Choice B incorrectly gives $9y\sqrt{3}$, which would result from factoring $27$ as $81 \times 3$.
13. Simplify: $\sqrt{128}$
    Answer: $8\sqrt{2}$ To simplify $\sqrt{128}$, we factor out perfect squares from under the radical. $\sqrt{128} = \sqrt{(64 \times 2)} = \sqrt{64} \times \sqrt{2} = 8\sqrt{2}$. The simplified form is $8\sqrt{2}$ since 64 is a perfect square. Choice C (16) would be if we incorrectly treated $\sqrt{128}$ as $\sqrt{256}$.
14. Simplify: $\sqrt{45}$
    Answer: $3\sqrt{5}$ To simplify $\sqrt{45}$, we factor out perfect squares from under the radical. $\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$. The simplified form is $3\sqrt{5}$ since 9 is a perfect square. Choice C (9) would be if we incorrectly treated $\sqrt{45}$ as $\sqrt{81}$.
15. Simplify: $\sqrt{50}$
    Answer: $5\sqrt{2}$ To simplify $\sqrt{50}$, we need to factor out perfect squares from under the radical. $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$. The simplified form is $5\sqrt{2}$ since 25 is a perfect square. Choice B ($\sqrt{25}$) equals 5, not the simplified radical form.
16. What is $8^0$?
    Answer: $1$ The zero exponent rule states that any non-zero number raised to the power of zero equals 1. $8^0 = 1$. This is a fundamental property of exponents that applies to all non-zero bases. Choice B incorrectly gives 0, confusing the exponent value with the result.
17. Simplify: $\sqrt{200}$
    Answer: $10\sqrt{2}$ To simplify $\sqrt{200}$, we factor out perfect squares from under the radical. $\sqrt{200} = \sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2} = 10\sqrt{2}$. The simplified form is $10\sqrt{2}$ since 100 is a perfect square. Choice C (20) would be if we incorrectly treated $\sqrt{200}$ as $\sqrt{400}$.
18. Simplify: $\sqrt{12}$
    Answer: $2\sqrt{3}$ To simplify $\sqrt{12}$, we factor out perfect squares from under the radical. $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$. The simplified form is $2\sqrt{3}$ since 4 is a perfect square. Choice C ($4$) would be if we incorrectly treated $\sqrt{12}$ as $\sqrt{16}$.
19. Which expression equals $f^{-6}$?
    Answer: $1/f^6$ The negative exponent rule states that $f^{-6} = 1/f^6$. A negative exponent means to take the reciprocal and make the exponent positive. The expression equals $1/f^6$. Choice B incorrectly gives $-f^6$, confusing the negative sign with the negative exponent rule.
20. Simplify: $\sqrt{18}$
    Answer: $3\sqrt{2}$ To simplify $\sqrt{18}$, we factor out perfect squares from under the radical. $\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$. The simplified form is $3\sqrt{2}$ since 9 is a perfect square. Choice C ($6$) would be if we incorrectly added 3 + 3 instead of multiplying.

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