Exponents and Roots - Quantitative Reasoning

1. What is the value of $\sqrt{49}$?

   $7$. Since $7^2 = 49$ and 7 is nonnegative, it is the principal square root.

2. What is the simplified form of $\sqrt{72}$?

   $6\sqrt{2}$. Factor out perfect square: $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$.

3. State the quotient of powers rule for the same base: $\frac{a^m}{a^n}$ equals what (for $a \ne 0$)?

   $a^{m-n}$. When dividing powers with the same base, subtract the exponents according to the quotient rule.

4. State the definition of a fractional exponent: $a^{\frac{1}{n}}$ equals what (for $a \ge 0$)?

   $\sqrt[n]{a}$. A fractional exponent with numerator 1 denotes the nth root of the base.

5. What is the simplified form of $\frac{\sqrt{18}}{\sqrt{2}}$ (with $\sqrt{\cdot}$ principal)?

   $3$. Rationalize by combining: $\sqrt{\frac{18}{2}} = \sqrt{9} = 3$.

6. State the power of a product rule: $(ab)^n$ equals what?

   $a^n b^n$. Raising a product to a power distributes the exponent to each factor according to the power of a product rule.

7. What is the simplified form of $\sqrt{\frac{50}{2}}$?

   $5$. Simplify fraction inside: $\frac{50}{2} = 25$, and $\sqrt{25} = 5$.

8. What is the simplified form of $\sqrt{x^2}$ in terms of $x$ for real $x$?

   $|x|$. The principal square root of a square yields the absolute value to ensure nonnegativity.

9. What is the value of $27^{\frac{2}{3}}$?

   $9$. $27 = 3^3$, so $27^{2/3} = (3^3)^{2/3} = 3^2 = 9$ using fractional exponent rules.

10. What is the value of $(2x^3)^2$?

    $4x^6$. Distribute the exponent: $(2)^2 = 4$ and $(x^3)^2 = x^6$.

11. What is the value of $2^3 \cdot 2^5$?

    $2^8$. Add the exponents when multiplying powers with the same base: $3 + 5 = 8$.

12. What is the simplified form of $\frac{a^{-3}b^2}{a^2 b^{-1}}$ (for $a,b \ne 0$)?

    $\frac{b^3}{a^5}$. Combine exponents for like bases: $a^{-3-2} = a^{-5}$ and $b^{2-(-1)} = b^3$, then apply negative exponent rule.

13. State the product of powers rule for the same base: $a^m \cdot a^n$ equals what?

    $a^{m+n}$. When multiplying powers with the same base, add the exponents according to the product rule.

14. State the power of a quotient rule: $\left(\frac{a}{b}\right)^n$ equals what (for $b \ne 0$)?

    $\frac{a^n}{b^n}$. Raising a quotient to a power distributes the exponent to both numerator and denominator according to the power of a quotient rule.

15. What is the value of $\sqrt{(-5)^2}$?

    $5$. $(-5)^2 = 25$, and the principal square root of 25 is 5.

16. What is the value of $\sqrt{\frac{1}{81}}$?

    $\frac{1}{9}$. The square root of a fraction is the quotient of the square roots, yielding $\frac{1}{9}$ since $\sqrt{1}=1$ and $\sqrt{81}=9$.

17. State the power of a power rule: $(a^m)^n$ equals what?

    $a^{mn}$. Raising a power to another power multiplies the exponents according to the power of a power rule.

18. State the conversion rule: $a^{\frac{m}{n}}$ equals what (for $a \ge 0$)?

    $\sqrt[n]{a^m}$. A fractional exponent $\frac{m}{n}$ means taking the nth root of the base raised to the m power.

19. What is the value of $(3^2)^4$?

    $3^8$. Multiply the exponents for a power raised to another: $2 \times 4 = 8$.

20. What is the value of $a^0$ for $a \ne 0$?

    $1$. Any non-zero base raised to the zero exponent equals 1 by definition of the zero exponent rule.

21. What is the simplified form of $\frac{x^6}{x^{-2}}$ (for $x \ne 0$)?

    $x^8$. Subtract exponents when dividing: $6 - (-2) = 8$.

22. What is the value of $\frac{5^7}{5^2}$?

    $5^5$. Subtract the exponents when dividing powers with the same base: $7 - 2 = 5$.

23. What is the value of $16^{\frac{3}{4}}$?

    $8$. $16 = 2^4$, so $16^{3/4} = (2^4)^{3/4} = 2^3 = 8$ using fractional exponent rules.

24. State the negative exponent rule: $a^{-n}$ equals what (for $a \ne 0$)?

    $\frac{1}{a^n}$. A negative exponent indicates the reciprocal of the base raised to the positive exponent.

25. State the principal square root convention: what does $\sqrt{a}$ mean for $a \ge 0$?

    The nonnegative number whose square is $a$. The principal square root is defined as the nonnegative number that, when squared, gives the original value.

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